tabla integrales
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17 TABLES of SPECIAL INDEFINITE INTEGRALS
Here we provide tables of special indefinite integrals. As stated in the remarks on page 67, here a, b, p, q, n are constants, restricted if indicated; e = 2.71828 . . . is the natural base of logarithms; ln u denotes the natural logarithm of u, where it is assumed that u > 0 (in general, to extend formulas to cases where u < 0 as well, replace ln u by ln |u|); all angles are in radians; and all constants of integration are omitted but implied. It is assumed in all cases that division by zero is excluded.
Our integrals are divided into types which involve the following algebraic expressions and functions:
(1) ax + b (13) ax bx c2 + + (25) eax
(2) ax b+ (14) x3 + a3 (26) ln x
(3) ax + b and px + q (15) x a4 4± (27) sinh ax
(4) ax b+ and px + q (16) x an n± (28) cosh ax
(5) ax b px q+ +and (17) sin ax (29) sinh ax and cosh ax
(6) x2 + a2 (18) cos ax (30) tanh ax
(7) x2 – a2, with x2 > a2 (19) sin ax and cos ax (31) coth ax
(8) a2 – x2, with x2 < a2 (20) tan ax (32) sech ax
(9) x a2 2+ (21) cot ax (33) csch ax
(10) x a2 2− (22) sec ax (34) inverse hyperbolic functions
(11) a x2 2− (23) csc ax
(12) ax2 + bx + c (24) inverse trigonometric functions
Some integrals contain the Bernouilli numbers Bn and the Euler numbers E
n defined in Chapter 23.
(1) Integrals Involving ax � b
17.1.1. dx
ax b aax b
+= +∫
1ln( )
17.1.2. x dx
ax bxa
ba
ax b+ = − +∫ 2 ln ( )
17.1.3.
x dxax b
ax ba
b ax ba
ba
ax b2 2
3 3
2
322
+= + − + + +( ) ( )
ln( )∫∫17.1.4.
dxx ax b b
xax b( )
ln+ = +⎛⎝⎜
⎞⎠⎟∫
1
17.1.5.
dxx ax b bx
ab
ax bx2 2
1( )
ln+ = − + +⎛⎝⎜
⎞⎠⎟∫
17.1.6.
dxax b a ax b( ) ( )+ = −
+∫ 2
1
17.1.7. x dx
ax bb
a ax b aax b
( ) ( )ln ( )+ = + + +∫ 2 2 2
1
17.1.8. x dx
ax bax b
ab
a ax bb
aax b
2
2 3
2
3 3
2( ) ( )
ln ( )+ = + − + − +∫
17.1.9. dxx ax b b ax b b
xax b( ) ( )
ln+ = + + +⎛⎝⎜
⎞⎠⎟∫ 2 2
1 1
71
72
17.1.10.
dxx ax b
ab ax b b x
ab
ax bx2 2 2 2 3
1 2( ) ( )
ln+ = −+ − + +⎛
⎝⎜⎞⎠⎠⎟∫
17.1.11.
dxax b ax b( ) ( )+ = −
+∫ 3 2
12
17.1.12.
x dxax b a ax b
ba ax b( ) ( ) ( )+ = −
+ + +∫ 3 2 2 2
12
17.1.13.
x dxax b
ba ax b
ba ax b a
a2
3 3
2
3 2 3
22
1( ) ( ) ( )
ln (+ = + − + + xx b+∫ )
17.1.14.
( )( )
( ). , . .ax b dx
ax bn a
nnn
+ = ++ = −
+1
11 17 1If see 11.∫
17.1.15.
x ax b dxax bn a
b ax bn
nn n
( )( )( )
( )(
+ = ++ − +
+∫+ +2
2
1
2 1)), ,
an2 1 2≠ − −
If n = –1, –2, see 17.1.2 and 17.1.7.
17.1.16.
x ax b dxax bn a
b ax bn
nn n
23
3
2
32
( )( )( )
( )(
+ = ++ − +
∫+ +
++ + ++
+
2 13
2 1
3)( )( )a
b ax bn a
n
If n = –1, –2, –3, see 17.1.3, 17.1.8, and 17.1.13.
17.1.17.
x ax b dx
x ax bm n
nbm n
x ax
m n
m nm
∫ + =
++ + + + + +
+
( )
( )(
1
1 1bb dx
x ax bm n a
mbm n a
x
n
m nm
)
( )( ) ( )
−
+−
∫+
+ + − + +
1
11
1 1(( )
( )( ) ( )
ax b dx
x ax bn b
m nn b
n
m n
+
− ++ + + +
+
∫+ +1 1
12
1xx ax b dxm n∫ +
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
+( ) 1
(2) Integrals Involving ax b�
17.2.1.
dx
ax b
ax ba+
= +∫2
17.2.2.
x dx
ax b
ax ba
ax b+
= − +∫2 2
3 2
( )
17.2.3.
x dx
ax b
a x abx ba
ax b2 2 2 2
3
2 3 4 815+
= − + +∫( )
17.2.4.dx
x ax b
b
ax b b
ax b b
b
ax b+=
+ −+ +
⎛
⎝⎜
⎞
⎠⎟
−+
−−
1
2 1
ln
tanbb
⎧
⎨
⎪⎪
⎩
⎪⎪
∫
17.2.5.dx
x ax b
ax bbx
ab
dx
x ax b2 2+= − + −
+∫∫ (see 17.2.12.)
17.2.6. ax b dxax b
a+ =
+∫
23
3( )
17.2.7. x ax b dxax b
aax b+ = − +∫
2 3 215 2
3( )( )
TABLES OF SPECIAL INDEFINITE INTEGRALS
17.2.8. x ax b dxa x abx b
aax b2
2 2 2
332 15 12 8
105+ = − + +∫
( )( )
17.2.9.ax b
xdx ax b b
dx
x ax b
+ = + ++∫∫ 2 (See 17.2.12.)
17.2.10.ax bx
dxax b
xa dx
x ax b
+ = − + ++∫ ∫2 2
(See 17.2.12.)
17.2.11.x
ax bdx
x ax bm a
mbm a
x
ax b
m m m
+= +
+ − + +∫−2
2 12
2 1
1
( ) ( )ddx∫
17.2.12.dx
x ax b
ax bm bx
m am b
dx
xm m m+= − +
− − −−−( )
( )( )12 32 21 −− +∫∫ 1 ax b
17.2.13. x ax b dxx
m aax b
mbm a
xmm
m+ = + + − +−2
2 32
2 33 2 1
( )( )
( )/ aax b dx+∫∫
17.2.14.ax bx
dxax b
m xa
mdx
x ax bm m m
+ = − +− + − +− −∫ ∫( ) ( )1 2 11 1
17.2.15.ax bx
dxax b
m bxm amm m
+ = − +− − −
−−( )
( )( )( )
/3 2
112 52 2 bb
ax bx
dxm
+−∫∫ 1
17.2.16. ( )( )
( )/
( )/
ax b dxax ba m
mm
+ = ++∫
+2
2 2
2
22
17.2.17. x ax b dxax ba m
b ax bmm
( )( )
( )(/
( )/
+ = ++ − +∫
+2
4 2
2
24
2 ))( )
( )/m
a m
+
+2 2
2 2
17.2.18. x ax b dxax ba m
b axmm
2 26 2
3
26
4( )
( )( )
(/( )/
+ = ++
− +∫+ bb
a mb ax b
a m
m m)( )
( )( )
( )/ ( )/+ +
++ +
+
4 2
3
2 2 2
342
2
17.2.19.( ) ( ) ( )/ / ( )/ax b
xdx
ax bm
bax b
xd
m m m+ = + + +∫ ∫
−2 2 2 22xx
17.2.20.( ) ( ) ( )/ ( )/ax b
xdx
ax bbx
mab
ax bm m m+ = − + + ++
∫2
2
2 2
2
//2
xdx∫
17.2.21.dx
x ax b m b ax b bdx
x axm m( ) ( ) ( ) (/ ( )/+ = − + + +−2 2 2
22
1bb m)( )/−∫∫ 2 2
(3) Integrals Involving ax � b and px � q
17.3.1.
dxax b px q bp aq
px qax b( )( )
ln+ + = −++
⎛⎝⎜
⎞⎠⎟∫
1
17.3.2.x dx
ax b px q bp aqba
ax bqp
px q( )( )
ln( ) ln( )+ +
=−
+ − +1 ⎧⎧⎨⎩
⎫⎬⎭∫
TABLES OF SPECIAL INDEFINITE INTEGRALS 73
74
17.3.3.
dxax b px q bp aq ax b
pbp aq
px qax( ) ( )
ln+ + = − + + −++2
1 1bb
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩
⎫⎬⎭∫
17.3.4.
x dxax b px q bp aq
qbp aq
ax bpx q( ) ( )
ln+ + = − −++
⎛⎝⎜
⎞2
1⎠⎠⎟ − +
⎧⎨⎩
⎫⎬⎭∫
ba ax b( )
17.3.5.x dx
ax b px qb
bp aq a ax b bp aq
2
2
2
2
1( ) ( ) ( ) ( ) (+ +
=− +
+− ))
ln ( )( )
ln ( )2
2
2
2qp
px qb bp aq
aax b+ + − +
⎧⎨⎩
⎫⎬⎭
∫
17.3.6.dx
ax b px q n bp aq ax b pm n m( ) ( ) ( )( ) ( ) (+ + = −− − + −
11
11 xx q
a m ndx
ax b px q
n
m n
+{+ + − + + }
−
−
∫
∫
)
( )( ) ( )
1
12
17.3.7.ax bpx q
dxaxp
bp aqp
px q++ = + − +∫ 2 ln( )
17.3.8.( )( )
( )( )( )(
ax bpx q
dx
n bp aqax bpx
m
n
m
++
=
−− −
+ +11
1
+++ − − +
+⎧⎨⎩
⎫⎬⎭
−
− −∫qn m a
ax bpx q
dxn
m
n)( )
( )( )1 1
2
1(( )
( )( )
( )( )
n m pax bpx q
m bp aqax bm
n
m
− −++
+ − +−
−
1 1
1
(( )
( )( )
( )
px qdx
n pax bpx q
m
n
m
n
+⎧⎨⎩
⎫⎬⎭
−−
++
−
∫
−
11 1
aaax bpx q
dxm
n
( )( )
++
⎧⎨⎩
⎫⎬⎭
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
−
−∫
∫1
1
(4) Integrals Involving ax b� and px � q
17.4.1.
px q
ax bdx
apx aq bpa
ax b++
= + − +∫2 3 2
3 2
( )
17.4.2.
dx
px q ax b
bp aq p
p ax b bp aq
p ax b
( )
ln( )
( )
+ += −
+ − −
+∫
1
++ −
⎛
⎝⎜
⎞
⎠⎟
−+
−
⎧
⎨
⎪⎪
⎩
⎪ −
bp aq
aq bp p
p ax baq bp
2 1tan( )
⎪⎪
17.4.3.
ax bpx q
dx
ax bp
bp aq
p p
p ax b bp aq
p a++
=
+ +− + − −2
ln( )
( xx b bp aq
ax bp
aq bp
p p
p ax b
+ + −
⎛
⎝⎜
⎞
⎠⎟
+ −− +−
)
tan(2 2
1 ))aq bp−
⎧
⎨
⎪⎪
⎩
⎪⎪
∫
17.4.4. ( )( )
( ) (px q ax b dx
px q ax bn p
bp aqnn
+ + = + ++ + −+2
2 3 2
1
nn ppx q
ax b
n
+++∫∫ 3)
( )
17.4.5.
dx
px q ax b
ax bn aq bp px q
nn n( ) ( )( )( )
(
+ += +
− − + +−12
1
−−− − + +∫ ∫ −
32 1 1
)( )( ) ( )
an aq bp
dx
px q ax bn
17.4.6.
( ) ( )( )
( )px q
ax bdx
px q ax bn a
n aq bpn n++
= + ++ + −2
2 12
(( )( )
2 1
1
n apx q dx
ax b
n
++
+
−
∫∫
17.4.7.
ax bpx q
dxax b
n p px qa
n pn n
++ = − +
− + + −−( ) ( ) ( ) ( )1 2 11∫∫ ∫ + +−
dx
px q ax bn( ) 1
TABLES OF SPECIAL INDEFINITE INTEGRALS
75
(5) Integrals Involving ax b� and px q+
17.5.1.dx
ax b px q
apa px q p ax b
ap
( )( )
ln ( ) ( )
t+ +=
+ + +( )
−
∫
2
2aan
( )( )
− − ++
⎧
⎨⎪⎪
⎩⎪⎪
1 p ax ba px q
17.5.2.x dx
ax b px q
ax b px qap
bp aqap
dx
( )( )
( )( )
(+ +=
+ +− +
∫ 2 aax b px q+ +∫ )( )
17.5.3. ( )( ) ( )( )(
ax b px q dxapx bp aq
apax b px q
b+ + = + + + + −24
pp aqap
dx
ax b px q
−+ +∫∫
)
( )( )
2
8
17.5.4.px qax b
dxax b px q
aaq bp
adx
ax b p
++ =
+ ++ −
+∫( )( )
( )(2 xx q+∫ )
17.5.5.dx
px q ax b px q
ax b
aq bp px q( ) ( )( ) ( )+ + += +
− +∫2
(6) Integrals Involving x2 � a2
17.6.1.dx
x a axa2 2
11+ = −∫ tan
17.6.2.x dx
x ax a
2 22 21
2+= +∫ ln ( )
17.6.3. x dxx a
x axa
2
2 21
+ = − −∫ tan
17.6.4. x dxx a
x ax a
3
2 2
2 22 2
2 2+ = − +∫ ln ( )
17.6.5. dxx x a a
xx a( )
ln2 2 2
2
2 2
12+
=+
⎛⎝⎜
⎞⎠⎟∫
17.6.6.dx
x x a a x axa2 2 2 2 3
11 1( )
tan+ = − − −∫
17.6.7.dx
x x a a x ax
x a3 2 2 2 2 4
2
2 2
12
12( )
ln+
= − −+
⎛⎝⎜
⎞⎠⎟∫
17.6.8.dx
x ax
a x a axa( ) ( )
tan2 2 2 2 2 2 31
21
2+ = + + −∫
17.6.9.x dx
x a x a( ) ( )2 2 2 2 2
12+ = −
+∫
17.6.10.x dx
x ax
x a axa
2
2 2 2 2 21
21
2( ) ( )tan+ = −
+ + −∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
76
17.6.11.x dx
x aa
x ax a
3
2 2 2
2
2 22 2
212( ) ( )
ln( )+
=+
+ +∫
17.6.12.dx
x x a a x a ax
x a( ) ( )ln2 2 2 2 2 2 4
2
2 2
12
12+ = + + +
⎛⎝⎜
⎞⎠⎟∫∫
17.6.13.dx
x x a a xx
a x a axa2 2 2 2 4 4 2 2 5
112
32( ) ( )
tan+ = − − + − −∫
17.6.14.dx
x x a a x a x a ax
x3 2 2 2 4 2 4 2 2 6
2
2
12
12
1( ) ( )
ln+ = − − + − + aa2
⎛⎝⎜
⎞⎠⎟∫
17.6.15.dx
x ax
n a x an
n an n( ) ( ) ( ) ( )2 2 2 2 2 1 22 12 3
2 2+ = − + + −−−∫∫ ∫ + −
dxx a n( )2 2 1
17.6.16.x dx
x a n x an n( ) ( )( )2 2 2 2 1
12 1+ = −
− + −∫
17.6.17.dx
x x a n a x a adx
x xn n( ) ( ) ( ) (2 2 2 2 2 1 2 2
12 1
1+ = − + + +∫ − aa n2 1) −∫
17.6.18.x dx
x ax dx
x aa
x dxx
m
n
m
n
m
( ) ( ) (2 2
2
2 2 12
2
2+ = + −−
−
−
∫∫ ++∫ a n2 )
17.6.19.
dxx x a a
dxx x a a
dxx xm n m n m( ) ( ) (2 2 2 2 2 1 2 2
1 1+ = + −∫ − − 22 2+∫∫ a n)
(7) Integrals Involving x2 � a2, x2 > a2
17.7.1.dx
x a ax ax a a
xa2 2
112
1− = −
+⎛⎝⎜
⎞⎠⎟ −∫ −ln cothor
17.7.2.x dx
x ax a2 2
2 212− = −∫ ln ( )
17.7.3.x dx
x ax
a x ax a
2
2 2 2− = + −+
⎛⎝⎜
⎞⎠⎟∫ ln
17.7.4.x dx
x ax a
x a3
2 2
2 22 2
2 2−= + −∫ ln( )
17.7.5.dx
x x a ax a
x( )ln2 2 2
2 2
2
12− = −⎛
⎝⎜⎞⎠⎟∫
17.7.6.dx
x x a a x ax ax a2 2 2 2 3
1 12( )
ln− = + −+
⎛⎝⎜
⎞⎠⎟∫
17.7.7.dx
x x a a x ax
x a3 2 2 2 2 4
2
2 2
12
12( )
ln− = − −⎛⎝⎜
⎞⎠⎟∫
17.7.8.dx
x ax
a x a ax ax a( ) ( )
ln2 2 2 2 2 2 321
4− = −− − −
+⎛⎝⎜
⎞⎠⎟∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
77
17.7.9. x dx
x a x a( ) ( )2 2 2 2 2
12− = −
−∫
17.7.10.x dx
x ax
x a ax ax a
2
2 2 2 2 221
4( ) ( )ln− = −
− + −+
⎛⎝⎜
⎞⎠⎟∫
17.7.11.x dx
x aa
x ax a
3
2 2 2
2
2 22 2
212( ) ( )
ln( )−
= −−
+ −∫
17.7.12.dx
x x a a x a ax
x a( ) ( )ln2 2 2 2 2 2 4
2
2 2
12
12− = −
− + −⎛⎝⎜
⎞⎠⎟⎟∫
17.7.13.dx
x x a a xx
a x a ax ax a2 2 2 2 4 4 2 2 5
12
34( ) ( )
ln− = − − − − −+
⎛⎛⎝⎜
⎞⎠⎟∫
17.7.14.dx
x x a a x a x a ax
x3 2 2 2 4 2 4 2 2 6
2
2
12
12
1( ) ( )
ln− = − − − + − aa2
⎛⎝⎜
⎞⎠⎟∫
17.7.15.dx
x ax
n a x an
n an n( ) ( ) ( ) ( )2 2 2 2 2 12 12 3
2 2− = −− − − −
−− 22 2 2 1∫ ∫ − −dx
x a n( )
17.7.16.x dx
x a n x an n( ) ( )( )2 2 2 2 1
12 1− = −
− − −∫
17.7.17. dxx x a n a x a a
dxx xn n( ) ( ) ( ) (2 2 2 2 2 1 2 2
12 1
1− = −
− − −−∫ −− −∫ a n2 1)
17.7.18.x dx
x ax dx
x aa
x dxx a
m
n
m
n
m
( ) ( ) (2 2
2
2 2 12
2
2− = − + −−
−
−
22 )n∫∫∫
17.7.19.dx
x x a adx
x x a adx
x x am n m n m( ) ( ) (2 2 2 2 2 2 2 2
1 1− = − − −− 22 1)n−∫∫∫
(8) Integrals Involving x2 � a2, x2 < a2
17.8.1.dx
a x aa xa x a
xa2 2
112
1− = +
−⎛⎝⎜
⎞⎠⎟∫ −ln tanhor
17.8.2.x dx
a xa x2 2
2 212− = − −∫ ln ( )
17.8.3.x dx
a xx
a a xa x
2
2 2 2− = − + +−
⎛⎝⎜
⎞⎠⎟∫ ln
17.8.4.x dx
a xx a
a x3
2 2
2 22 2
2 2−= − − −∫ ln( )
17.8.5.dx
x a x ax
a x( )ln2 2 2
2
2 2
12− = −
⎛⎝⎜
⎞⎠⎟∫
17.8.6.dx
x a x a x aa xa x2 2 2 2 3
1 12( )
ln− = − + +−
⎛⎝⎜
⎞⎠⎟∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
78
17.8.7. dx
x a x a x ax
a x3 2 2 2 2 4
2
2 2
12
12( )
ln− = − + −⎛⎝⎜
⎞⎠⎟∫
17.8.8. dx
a xx
a a x aa xa x( ) ( )
ln2 2 2 2 2 2 321
4− = − + +−
⎛⎝⎜
⎞⎠⎟∫
17.8.9. x dx
a x a x( ) ( )2 2 2 2 2
12− = −∫
17.8.10.x dx
a xx
a x aa xa x
2
2 2 2 2 221
4( ) ( )ln− = − − +
−⎛⎝⎜
⎞⎠⎟∫
17.8.11.x dx
a xa
a xa x
3
2 2 2
2
2 22 2
212( ) ( )
ln( )−
=−
+ −∫
17.8.12.dx
x a x a a x ax
a x( ) ( )ln2 2 2 2 2 2 4
2
2 2
12
12− = − + −
⎛⎝⎜
⎞⎠⎟∫∫
17.8.13.dx
x a x a xx
a a x aa xa x2 2 2 2 4 4 2 2 5
12
34( ) ( )
ln− = − + − + +−
⎛⎛⎝⎜
⎞⎠⎟∫
17.8.14.dx
x a x a x a a x ax
a3 2 2 2 4 2 4 2 2 6
2
2
12
12
1( ) ( )
ln− = − + − + − xx2
⎛⎝⎜
⎞⎠⎟∫
17.8.15.dx
a xx
n a a xn
n an n( ) ( ) ( ) ( )2 2 2 2 2 1 22 12 3
2 2− = − − + −−−∫∫ ∫ − −
dxa x n( )2 2 1
17.8.16.x dx
a x n a xn n( ) ( )( )2 2 2 2 1
12 1− = − − −∫
17.8.17.dx
x a x n a a x adx
x a xn n( ) ( ) ( ) (2 2 2 2 2 1 2 2
12 1
1− = − − + −− 22 1)n−∫∫
17.8.18.x dx
a xa
x dxa x
x dxa x
m
n
m
n
m
( ) ( ) (2 22
2
2 2
2
2− = − − −∫ ∫− −
22 1)n−∫
17.8.19.dx
x a x adx
x a x adx
x am n m n m( ) ( ) (2 2 2 2 2 1 2 2
1 1− = − +− −∫ 22 2−∫∫ x n)
(9) Integrals Involving x a2 2+
17.9.1. dx
x ax x a
xa2 2
2 2 1
+= + +∫ −ln( ) sinhor
17.9.2.x dx
x ax a
2 2
2 2
+= +∫
17.9.3.x dx
x a
x x a ax x a
2
2 2
2 2 22 2
2 2+= + − + +∫ ln( )
17.9.4.x dx
x a
x aa x a
3
2 2
2 2 3 22 2 2
3+= + − +∫
( ) /
TABLES OF SPECIAL INDEFINITE INTEGRALS
79
17.9.5. dx
x x a aa x a
x2 2
2 21
+= − + +⎛
⎝⎜⎞
⎠⎟∫ ln
17.9.6. dx
x x a
x aa x2 2 2
2 2
2+= − +
∫
17.9.7. dx
x x a
x aa x a
a x ax3 2 2
2 2
2 2 3
2 2
21
2+= − + + + +⎛
⎝⎜⎞
⎠⎟∫ ln
17.9.8. x a dxx x a a
x x a2 22 2 2
2 2
2 2+ = + + + +∫ ln( )
17.9.9. x x a dxx a2 2
2 2 3 2
3+ = +∫
( ) /
17.9.10. x x a dxx x a a x x a a
x x2 2 22 2 3 2 2 2 2 4
2
4 8 8+ = + − + − +( )
ln(/
++∫ a2 )
17.9.11. x x a dxx a a x a3 2 2
2 2 5 2 2 2 2 3 2
5 3+ = + − +∫
( ) ( )/ /
17.9.12. x ax
dx x a aa x a
x
2 22 2
2 2+ = + − + +⎛
⎝⎜⎞
⎠⎟∫ ln
17.9.13.x a
xdx
x ax
x x a2 2
2
2 22 2+ = − + + + +∫ ln( )
17.9.14.x a
xdx
x ax a
a x ax
2 2
3
2 2
2
2 2
21
2+ = − + − + +⎛
⎝⎜⎞
⎠⎟∫ ln
17.9.15.dx
x ax
a x a( ) /2 2 3 2 2 2 2+ =+∫
17.9.16.x dx
x a x a( ) /2 2 3 2 2 2
1+ = −
+∫
17.9.17.x dx
x ax
x ax x a
2
2 2 3 2 2 2
2 2
( )ln ( )/+ = −
++ + +∫
17.9.18.x dx
x ax a
a
x a
3
2 2 3 22 2
2
2 2( ) /+ = + ++∫
17.9.19.dx
x x a a x a aa x a
x( )ln/2 2 3 2 2 2 2 3
2 21 1+ =
+− + +⎛
⎝⎜⎞
⎠⎟∫
17.9.20.dx
x x ax aa x
x
a x a2 2 2 3 2
2 2
4 4 2 2( ) /+ = − + −+∫
17.9.21.dx
x x a a x x a a x a a3 2 2 3 2 2 2 2 2 4 2 2 5
1
2
3
2
32( )
l/+ = −+
−+
+ nna x a
x+ +⎛
⎝⎜⎞
⎠⎟∫2 2
17.9.22. ( )( )
l//
x a dxx x a a x x a
a2 2 3 22 2 3 2 2 2 2
4
43
838
+ = + + + +∫ nn( )x x a+ +2 2
17.9.23. x x a dxx a
( )( )/
/2 2 3 2
2 2 5 2
5+ = +∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
80
17.9.24. x x a dxx x a a x x a2 2 2 3 2
2 2 5 2 2 2 2 3 2
6 2( )
( ) ( )// /
+ = + − +∫ 44 16 16
4 2 2 62 2− + − + +a x x a a
x x aln ( )
17.9.25. x x a dxx a a x a3 2 2 3 2
2 2 7 2 2 2 2 5 2
7 5( )
( ) ( )// /
+ = + − +∫
17.9.26.( ) ( )
ln/ /x a
xdx
x aa x a a
a x2 2 3 2 2 2 3 22 2 2 3
2
3+ = + + + − + + aa
x
2⎛
⎝⎜⎞
⎠⎟∫
17.9.27.( ) ( )
ln/ /x a
xdx
x ax
x x aa
2 2 3 2
2
2 2 3 2 2 223
232
+ = − + + + + (( )x x a+ +∫ 2 2
17.9.28.( ) ( )
ln/ /x a
xdx
x ax
x a a2 2 3 2
3
2 2 3 2
22 2
232
32
+ = − + + + − aa x ax
+ +⎛
⎝⎜⎞
⎠⎟∫2 2
(10) Integrals Involving x a2 2−
17.10.1.dx
x ax x a
x dx
x ax a
2 2
2 2
2 2
2 2
−= + −
−= −∫ ∫ln( ),
17.10.2.x dx
x a
x x a ax x a
2
2 2
2 2 22 2
2 2−= − + + −∫ ln ( )
17.10.3.x dx
x a
x aa x a
3
2 2
2 2 3 22 2 2
3−= − + −∫
( ) /
17.10.4.dx
x x a axa2 2
11
−= −∫ sec
17.10.5.dx
x x a
x aa x2 2 2
2 2
2−= −
∫
17.10.6.dx
x x a
x aa x a
xa3 2 2
2 2
2 2 31
21
2−= − + −∫ sec
17.10.7. x a dxx x a a
x x a2 22 2 2
2 2
2 2− = − − + −∫ ln ( )
17.10.8. x x a dxx a2 2
2 2 3 2
3− = −∫
( ) /
17.10.9. x x a dxx x a a x x a a
x x2 2 22 2 3 2 2 2 2 4
2
4 8 8− = − + − − +( )
ln (/
−−∫ a2 )
17.10.10. x x a dxx a a x a3 2 2
2 2 5 2 2 2 2 3 2
5 3− = − + −∫
( ) ( )/ /
17.10.11.x a
xdx x a a
xa
2 22 2 1− = − − −∫ sec
17.10.12.x a
xdx
x ax
x x a2 2
2
2 22 2− = − − + + −∫ ln ( )
TABLES OF SPECIAL INDEFINITE INTEGRALS
81
17.10.13.x a
xdx
x ax a
xa
2 2
3
2 2
21
21
2− = − − + −∫ sec
17.10.14.dx
x ax
a x a( ) /2 2 3 2 2 2 2− = −−∫
17.10.15.x dx
x a x a( ) /2 2 3 2 2 2
1− = −
−∫
17.10.16.x dx
x ax
x ax x a
2
2 2 3 2 2 2
2 2
( )ln( )
/−= −
−+ + −∫
17.10.17. x dxx a
x aa
x a
3
2 2 3 22 2
2
2 2( ) /− = − −−∫
17.10.18.dx
x x a a x a axa( )
sec/2 2 3 2 2 2 2 3
11 1−
= −−
− −∫
17.10.19.dx
x x ax aa x
x
a x a2 2 2 3 2
2 2
4 4 2 2( ) /− = − − −−∫
17.10.20.dx
x x a a x x a a x a a3 2 2 3 2 2 2 2 2 4 2 2 5
1
2
3
2
32( )
se/−
=−
−−
− cc−∫ 1 xa
17.10.21. ( )( )
l//
x a dxx x a a x x a
a2 2 3 22 2 3 2 2 2 2
4
43
838
− = − − − +∫ nn( )x x a+ −2 2
17.10.22. x x a dxx a
( )( )/
/2 2 3 2
2 2 5 2
5− = −∫
17.10.23. x x a dxx x a a x x a2 2 2 3 2
2 2 5 2 2 2 2 3 2
6 2( )
( ) ( )// /
− = − + −∫ 44 16 16
4 2 2 62 2− − + + −a x x a a
x x aln ( )
17.10.24. x x a dxx a a x a3 2 2 3 2
2 2 7 2 2 2 2 5 2
7 5( )
( ) ( )// /
− = − + −∫
17.10.25.( ) ( )
sec/ /x a
xdx
x aa x a a
x2 2 3 2 2 2 3 22 2 2 3 1
3− = − − − +∫ −
aa
17.10.26.( ) ( )
l/ /x a
xdx
x ax
x x aa
2 2 3 2
2
2 2 3 2 2 223
232
− = − − + − −∫ nn ( )x x a+ −2 2
17.10.27.( ) ( )
se/ /x a
xdx
x ax
x aa
2 2 3 2
3
2 2 3 2
2
2 2
23
232
− = − − + − − cc−∫ 1 xa
(11) Integrals Involving a x2 2−
17.11.1.dx
a x
xa2 2
1
−= −∫ sin
17.11.2.x dx
a xa x
2 2
2 2
−= − −∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
82
17.11.3. x dx
a x
x a x a xa
2
2 2
2 2 21
2 2−= − − +∫ −sin
17.11.4. x dx
a x
a xa a x
3
2 2
2 2 3 22 2 2
3−= − − −∫
( ) /
17.11.5. dx
x a x aa a x
x2 2
2 21
−= − + −⎛
⎝⎜⎞
⎠⎟∫ ln
17.11.6. dx
x a x
a xa x2 2 2
2 2
2−= − −
∫
17.11.7. dx
x a x
a xa x a
a a xx3 2 2
2 2
2 2 3
2 2
21
2−= − − − + −⎛
⎝⎜⎞
⎠⎟∫ ln
17.11.8. a x dxx a x a x
a2 2
2 2 21
2 2− = − +∫ −sin
17.11.9. x a x dxa x2 2
2 2 3 2
3− = − −∫
( ) /
17.11.10. x a x dxx a x a x a x a2 2 2
2 2 3 2 2 2 2 41
4 8 8− = − − + − +∫ −( )
sin/ xx
a
17.11.11. x a x dxa x a a x3 2 2
2 2 5 2 2 2 2 3 2
5 3− = − − −∫
( ) ( )/ /
17.11.12.a x
xdx a x a
a a xx
2 22 2
2 2− = − − + −⎛
⎝⎜⎞
⎠⎟∫ ln
17.11.13.a x
xdx
a xx
xa
2 2
2
2 21− = − − − −∫ sin
17.11.14.a x
xdx
a xx a
a a xx
2 2
3
2 2
2
2 2
21
2− = − − + + −⎛
⎝⎜⎞
⎠⎟∫ ln
17.11.15.dx
a xx
a a x( ) /2 2 3 2 2 2 2− =−∫
17.11.16.x dx
a x a x( ) /2 2 3 2 2 2
1− =
−∫
17.11.17.x dx
a xx
a x
xa
2
2 2 3 2 2 2
1
( )sin/− =
−−∫ −
17.11.18.x dx
a xa x
a
a x
3
2 2 3 22 2
2
2 2( ) /− = − +−∫
17.11.19.dx
x a x a a x aa a x
x( )ln/2 2 3 2 2 2 2 3
2 21 1− =
−− + −⎛
⎝⎜⎞
⎠⎟∫
17.11.20.dx
x a xa xa x
x
a a x2 2 2 3 2
2 2
4 4 2 2( ) /− = − − +−∫
17.11.21.dx
x a x a x a x a a x a3 2 2 3 2 2 2 2 2 4 2 2 5
1
2
3
2
32( ) /− = −
−+
−−∫ lln
a a xx
+ −⎛
⎝⎜⎞
⎠⎟2 2
TABLES OF SPECIAL INDEFINITE INTEGRALS
83
17.11.22. ( )( )
s//
a x dxx a x a x a x
a2 2 3 22 2 3 2 2 2 2
4
43
838
− = − + − +∫ iin−1 xa
17.11.23. x a x dxa x
( )( )/
/2 2 3 2
2 2 5 2
5− = − −∫
17.11.24. x a x dxx a x a x a x2 2 2 3 2
2 2 5 2 2 2 2 3 2
6( )
( ) ( )// /
− = − − + −∫ 224 16 16
4 2 2 61+ − + −a x a x a x
asin
17.11.25. x a x dxa x a a x3 2 2 3 2
2 2 7 2 2 2 2 5 2
7 5( )
( ) ( )// /
− = − − −∫
17.11.26.( ) ( )
ln/ /a x
xdx
a xa a x a
a a2 2 3 2 2 2 3 22 2 2 3
2
3− = − + − − +
∫−−⎛
⎝⎜
⎞
⎠⎟
xx
2
17.11.27.( ) ( )
s/ /a x
xdx
a xx
x a xa
2 2 3 2
2
2 2 3 2 2 223
232
− = − − − − −∫ iin−1 xa
17.11.28.( ) ( )
l/ /a x
xdx
a xx
a xa
2 2 3 2
3
2 2 3 2
2
2 2
23
232
− = − − − − +∫ nna a x
x+ −⎛
⎝⎜⎞
⎠⎟2 2
(12) Integrals Involving ax2 � bx � c
17.12.1.dx
ax bx c
ac b
ax b
ac b
b ac
2
2
1
2
2
2
4
2
4
1
4
+ +=
−+−
−
−tan
ln22 4
2 4
2
2
ax b b ac
ax b b ac
+ − −+ + −
⎛
⎝⎜
⎞
⎠⎟
⎧
⎨⎪⎪
⎩⎪⎪
∫
If b ac ax bx c a x b a2 2 24 2= + + = +, ( / ) and the results 17.1.6 to 17.1.10 and 17.1.14 to 17.1.17 can be used. If b = 0 use results on page 75. If a or c = 0 use results on pages 71–72.
17.12.2.x dx
ax bx c aax bx c
ba
dxax bx c2
22
12 2+ + = + + − + +∫∫ ln ( )
17.12.3.x dx
ax bx cxa
ba
ax bx cb ac
a
2
2 22
2
222
2+ + = − + + + −∫ ln ( )
ddxax bx c2 + +∫
17.12.4.x dx
ax bx cx
m aca
x dxax bx c
ba
m m m
2
1 2
21+ + = − − + + −− −
( )xx dx
ax bx c
m−
+ +∫∫∫1
2
17.12.5.dx
x ax bx c cx
ax bx cbc
dxa( )
ln2
2
2
12 2+ + = + +
⎛⎝⎜
⎞⎠⎟
−∫ xx bx c2 + +∫
17.12.6.dx
x ax bx cbc
ax bx cx cx2 2 2
2
221
( )ln+ + = + +⎛
⎝⎜⎞⎠⎟
− +∫bb ac
cdx
ax bx c
2
2 2
22−
+ +∫
17.12.7.dx
x ax bx c n cxbc
dxx ax bxn n n( ) ( ) (2 1 1 2
11+ + = − − − +− − ++ − + +−∫∫∫ c
ac
dxx ax bx cn) ( )2 2
17.12.8.dx
ax bx cax b
ac b ax bx ca
ac( ) ( )( )2 2 2 2
24
24+ + = +
− + + + −− + +∫∫ bdx
ax bx c2 2
17.12.9.x dx
ax bx cbx c
ac b ax bx cb
a( ) ( )( )2 2 2 2
24 4+ + = − +
− + + −cc b
dxax bx c− + +∫ ∫2 2
TABLES OF SPECIAL INDEFINITE INTEGRALS
84
17.12.10.x dx
ax bx cb ac x bc
a ac b ax b
2
2 2
2
2 2
24( )
( )( )(+ + = − +
− + xx cc
ac bdx
ax bx c+ + − + +∫ ∫)2
4 2 2
17.12.11.x dx
ax bx cx
n m a ax bx c
m
n
m
n( ) ( ) ( )2
1
2 12 1+ + = − − − + +−
− ++ −− − + +
− −
−
∫∫( )
( ) ( )
( )
m cn m a
x dxax bx c
n m b
m
n
12 1
2
2
(( ) ( )2 1
1
2n m ax dx
ax bx c
m
n− − + +−
∫
17.12.12.x dx
ax bx c ax dx
ax bx ccn
n
n
n
2 1
2
2 3
2 1
1− −
−+ + = + + −( ) ( ) aa
x dxax bx c
ba
x dxax bx c
n
n
n
n∫ ∫− −
+ + − + +2 3
2
2 2
2( ) ( )∫∫∫
17.12.13.dx
x ax bx c c ax bx cbc
dxax bx( ) ( ) (2 2 2 2
12 2+ + = + + − + +∫ cc c
dxx ax bx c) ( )2 2
1+ + +∫∫
17.12.14.dx
x ax bx c cx ax bx cac
dxax b2 2 2 2 2
1 3( ) ( ) (+ + = − + + − +∫ xx c
bc
dxx ax bx c+ − + +∫∫ ) ( )2 2 2
2
17.12.15.dx
x ax bx c m cx ax bx cm n m n( ) ( ) ( )(
2 1 2 1
11+ + = − − + + −− −
mm n am c
dxx ax bx c
m n b
m n
+ −− + +
− + −
∫ ∫ −2 3
1
2
2 2
)( ) ( )
( )(( ) ( )m c
dxx ax bx cm n− + +−∫1 1 2
(13) Integrals Involving ax bx c2 � �
In the following results if b ac ax bx c a x b a2 24 2= + + = +, ( / ) and the results 17.1 can be used. If b = 0 use the results 17.9. If a = 0 or c = 0 use the results 17.2 and 17.5.
17.13.1.dx
ax bx c
aa ax bx c ax b
a
2
2
1
12 2
1 2+ +=
+ + + +
−−
−
ln ( )
sinaax b
b ac a
ax b
ac b
+−
⎛⎝⎜
⎞⎠⎟
+−
⎛⎝⎜
⎞⎠
−2
1
24
1 2
4or sinh ⎟⎟
⎧
⎨⎪⎪
⎩⎪⎪
∫
17.13.2.x dx
ax bx c
ax bx ca
ba
dx
ax bx c2
2
22+ += + + −
+ +∫ ∫
17.13.3.x dx
ax bx c
ax ba
ax bx cb ac
ad2
2 22
2
2
2 34
3 48+ +
= − + + + −∫
xx
ax bx c2 + +∫
17.13.4.dx
x ax bx c
c
c ax bx c bx cx
2
21 2 2
1+ +=
− + + + +⎛
⎝⎜⎞
⎠⎟
−
∫ln
cc
bx c
x b ac c
bx csin
| |sinh− −+
−⎛⎝⎜
⎞⎠⎟
− +1
2
12
4
1 2or
|| |x ac b4 2−⎛⎝⎜
⎞⎠⎟
⎧
⎨⎪⎪
⎩⎪⎪
17.13.5.dx
x ax bx c
ax bx ccx
bc
dx
x ax bx c2 2
2
22+ += − + + −
+ +∫ ∫
17.13.6. ax bx c dxax b ax bx c
aac b
adx
ax2
2 2
2
24
48
+ + = + + + + −∫( )
++ +∫ bx c
TABLES OF SPECIAL INDEFINITE INTEGRALS
85
17.13.7. x ax bx c dxax bx c
ab ax b
aax2
2 3 2
22
328
+ + = + + − + +( ) ( )/
bbx c
b ac ba
dx
ax bx c
+
− −+ +
∫
∫( )4
16
2
2 2
17.13.8. x ax bx c dxax b
aax bx c
b a2 22
2 3 226 5
245 4+ + = − + + + −
( ) / cca
ax bx c dx16 2
2∫ ∫ + +
17.13.9.ax bx c
xdx ax bx c
b dx
ax bx cc
dx
x ax
22
2 22+ + = + + +
+ ++∫ ∫ ++ +∫
bx c
17.13.10.ax bx c
xdx
ax bx cx
adx
ax bx c
b dx
x a
2
2
2
2 2+ + = − + + +
+ ++∫
xx bx c2 + +∫∫
17.13.11.dx
ax bx cax b
ac b ax bx c( )( )
( )/2 3 2 2 2
2 2
4+ + = +− + +∫
17.13.12.x dx
ax bx cbx c
b ac ax bx c( )( )
( )/2 3 2 2 2
2 2
4+ + = +− + +∫
17.13.13.x dx
ax bx cb ac x bc
a ac b ax
2
2 3 2
2
2
2 4 2
4( )( )
( )/+ + = − +
− 22 2
1
+ ++
+ +∫∫ bx c adx
ax bx c
17.13.14.dx
x ax bx c c ax bx c cdx
x ax bx c
b( ) /2 3 2 2 2
1 1+ + =
+ ++
+ +−
22 2 3 2cdx
ax bx c( ) /+ +∫∫∫
17.13.15.dx
x ax bx cax bx c
c x ax bx c
b2 2 3 2
2
2 2
22( ) /+ + = − + +
+ ++∫
−−+ +
−+ +
∫
∫
22
32
2 2 3 2
2 2
acc
dxax bx c
bc
dx
x ax bx c
( ) /
17.13.16. ( )( )( )/
/
ax bx c dxax b ax bx c
an
n2 1 2
2 1 224
+ + = + + +++
∫ (( )( )( )
( )( ) /
nn ac b
a nax bx c n
++ + −
++ + −
12 1 4
8 1
22 1 2 ddx∫
17.13.17. x ax bx c dxax bx c
a nn
n
( )( )
( )/
/2 1 2
2 3 2
2 3+ + = + +
+−+
+
∫bba
ax bx c dxn
22 1 2( ) /+ + +∫
17.13.18.dx
ax bx cax b
n ac b axn( )( )
( )( )(/2 1 2 2
2 22 1 4+ + = +
− −+ 22 1 2
2 2
8 12 1 4
+ +
+ −− − +
−∫ bx c
a nn ac b
dxax
n)
( )( )( ) (
/
bbx c n+ −∫ ) /1 2
17.13.19.dx
x ax bx c n c ax bx cn n( ) ( ) ( )/ /2 1 2 2 1 2
12 1+ + = − + ++ −∫
++ + + − + +− +∫1
22 1 2 2 1cdx
x ax bx cbc
dxax bx cn n( ) ( )/ /22∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
86
(14) Integrals Involving x3 � a3
Note that for formulas involving x3 – a3 replace a with –a.
17.14.1. dx
x a ax a
x ax a a3 3 2
2
2 2 2
16
1
3+= +
− +⎛⎝⎜
⎞⎠⎟ + −ln
( )tan 11 2
3
x a
a
−∫
17.14.2.x dx
x a ax ax a
x a a3 3
2 2
211
61
3+= − +
+⎛⎝⎜
⎞⎠⎟
+ −ln( )
tan22
3
x a
a
−∫
17.14.3.x dx
x ax a
2
3 33 31
3+ = +∫ ln ( )
17.14.4. dx
x x a ax
x a( )ln3 3 3
3
3 3
13+ = +
⎛⎝⎜
⎞⎠⎟∫
17.14.5. dx
x x a a x ax ax a
x a2 3 3 3 4
2 2
2
1 16( )
ln( )+
= − − − ++
⎛⎝⎜
⎞⎠⎟⎟
− −∫ −1
3
2
34
1
a
x a
atan
17.14.6. dx
x ax
a x a ax a
x ax a( ) ( )ln
( )3 3 2 3 3 3 5
2
2 231
9+=
++ +
− +⎛⎛⎝⎜
⎞⎠⎟ + −−∫
2
3 3
2
35
1
a
x a
atan
17.14.7. x dx
x ax
a x a ax ax a
x( ) ( )ln
(3 3 2
2
3 3 3 4
2 2
31
18+=
++ − +
+ aa a
x a
a)tan
2 4
11
3 3
2
3
⎛⎝⎜
⎞⎠⎟
+ −−∫
17.14.8. x dx
x a x a
2
3 3 2 3 3
13( ) ( )+ = − +∫
17.14.9. dx
x x a a x a ax
x a( ) ( )ln3 3 2 3 3 3 6
3
3 3
13
13+ = + + +
⎛⎝⎜
⎞⎠⎟∫∫
17.14.10.dx
x x a a xx
a x a ax dx
x2 3 3 2 6
2
6 3 3 6 3
13
43( ) ( )+
= − −+
−+∫ aa3
(See 17.14.2.)∫
17.14.11.x dx
x axm
ax dxx a
m m m
3 3
23
3
3 32+=
−−
+
− −
∫ ∫
17.14.12.dx
x x a a n x adx
x x an n n( ) ( ) ( )3 3 3 1 3 3 3 3
11
1+ = −
− − +− −∫ ∫∫
(15) Integrals Involving x4 � a4
17.15.1.dx
x a a
x ax a
x ax a4 4 3
2 2
2 2
1
4 2
2
2
1
2+ = + +− +
⎛⎝⎜
⎞⎠⎟
−∫ lnaa
xa
xa3
1 1
21
21
2tan tan− −−
⎛⎝⎜
⎞⎠⎟
− +⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥⎥
17.15.2.x dx
x a axa4 4 2
12
2
12+ = −∫ tan
17.15.3.x dx
x a a
x ax a
x ax a
2
4 4
2 2
2 2
1
4 2
2
2
1+ = − +
+ +⎛⎝⎜
⎞⎠⎟
−∫ ln22 2
12
121 1
a
xa
xa
tan tan− −−⎛⎝⎜
⎞⎠⎟
− +⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥⎥
TABLES OF SPECIAL INDEFINITE INTEGRALS
87
17.15.4. x dx
x ax a
3
4 44 41
4+ = +∫ ln ( )
17.15.5. dx
x x a ax
x a( )ln4 4 4
4
4 4
14+ = +
⎛⎝⎜
⎞⎠⎟∫
17.15.6.dx
x x a a x a
x ax a
x ax a2 4 4 4 5
2 2
2 2
1 1
4 2
2
2( )ln+ = − − − +
+ +⎛⎛⎝⎜
⎞⎠⎟
+ −⎛⎝⎜
⎞⎠⎟
− +⎛⎝
∫
− −1
2 21
21
25
1 1
a
xa
xa
tan tan ⎜⎜⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
17.15.7. dx
x x a a x axa3 4 4 4 2 6
12
2
12
12( )
tan+ = − − −∫
17.15.8. dx
x a ax ax a a
xa4 4 3 3
114
12− = −
+⎛⎝⎜
⎞⎠⎟ −∫ −ln tan
17.15.9. x dx
x a ax ax a4 4 2
2 2
2 2
14− = −
+⎛⎝⎜
⎞⎠⎟∫ ln
17.15.10.x dx
x a ax ax a a
xa
2
4 411
41
2− = −+
⎛⎝⎜
⎞⎠⎟ +∫ −ln tan
17.15.11.x dx
x ax a
3
4 44 41
4−= −∫ ln( )
17.15.12.dx
x x a ax a
x( )ln4 4 4
4 4
4
14− = −⎛
⎝⎜⎞⎠⎟∫
17.15.13.dx
x x a a x ax ax a a3 4 4 4 5 5
1 14
12( )
ln tan− = + −+
⎛⎝⎜
⎞⎠⎟ + −−∫ 1 x
a
17.15.14.dx
x x a a x ax ax a3 4 4 4 2 6
2 2
2 2
12
14( )
ln− = + −+
⎛⎝⎜
⎞⎠⎟∫
(16) Integrals Involving xn � an
17.16.1.dx
x x a nax
x an n n
n
n n( )ln
+=
+⎛⎝⎜
⎞⎠⎟∫
1
17.16.2.x dxx a n
x an
n nn n
−
+ = +∫1 1
ln ( )
17.16.3.x dx
x ax dx
x aa
x dxx
m
n n r
m n
n n rn
m n
n( ) ( ) (+ = + − +−
−
−
∫ 1 aan r)∫∫
17.16.4.dx
x x a adx
x x a adx
x xm n n r n m n n r n m n( ) ( ) (+ = + −− −∫1 1
1 nn n ra+∫∫ )
17.16.5.dx
x x a n a
x a a
x a an n n
n n n
n n n+= + −
+ +⎛
⎝⎜⎞
⎠⎟∫1
ln
17.16.6.dx
x x a nax a
xn n n
n n
n( )ln− = −⎛
⎝⎜⎞⎠⎟∫
1
TABLES OF SPECIAL INDEFINITE INTEGRALS
88
17.16.7. x dxx a n
x an
n nn n
−
−= −∫
1 1ln( )
17.16.8.x dx
x aa
x dxx a
x dxx a
m
n n rn
m n
n n r
m n
n n( ) ( ) ( )− = − + −− −
rr−∫∫∫ 1
17.16.9. dx
x x a adx
x x a adx
x x am n n r n m n n n r n m n( ) ( ) (−=
−−
−−
1 1nn r) −∫∫∫
17.16.10.dx
x x a n a
axn n n
n
n−= −∫
2 1cos
17.16.11.x dx
x a mak p
mx ap
m m m p
−
−−
+= − +∫
1
2 2 211 2 1
2sin
( )tan
π ccos[( ) / ]sin[( ) / ]
2 1 22 1 21
k ma k mk
m −−
⎛⎝⎜
⎞⎠⎟=
∑ ππ
−− − + −−
=∑1
22 1
22
22
1
2
mak p
mx ax
km p
k
m
cos( )
cos(π
ln11
22)π
ma+
⎛⎝⎜
⎞⎠⎟
where 0 < p � 2m.
17.16.12.x dx
x a makpm
x axkm
p
m m m p
−
−−= −
1
2 2 221
22cos ln cos
π π ++⎛⎝⎜
⎞⎠⎟
− −
=
−
−−
∑∫ a
makpm
x a
k
m
m p
2
1
1
211
sin tancoπ ss( / )
sin( / )
{ln
k ma k m
ma
k
m
m p
ππ
⎛⎝⎜
⎞⎠⎟
+
=
−
−
∑1
1
2
12
(( ) ( ) ln( )}x a x ap− + − +1
where 0 < p � 2m.
17.16.13.x dx
x a m a
p
m m
p
m p
−
+ +
−
− ++= −
+
1
2 1 2 1
1
2 1
2 12 1
2( )( )
sinkkpm
x a k ma k m
π ππ2 12 2 1
2 21
++ +−tan
cos[ /( )]sin[ /( ++
⎛⎝⎜
⎞⎠⎟
− −+
=
−
− +
∑∫ 1
12 1
2
1
1
2 1
)]
( )( )
cos
k
m
p
m pm akppm
x axk
ma
k
m π π2 1
22
2 1
1
2 2
1 ++
++
⎛⎝⎜
⎞⎠⎟
+ −
=∑ ln cos
( )) ln( )( )
p
m p
x am a
−
− +
++
1
2 12 1
where 0 < p � 2m + 1.
17.16.14. x dxx a m a
kpm
p
m m m p
−
+ + − +−= −
+ +
1
2 1 2 1 2 1
22 1
22( )
sinπ11
2 2 12 2 1
1tancos[ /( )]
sin[ /( )]− − +
+⎛⎝
x a k ma k m
ππ⎜⎜
⎞⎠⎟
++ +
−
=
− +
∑∫k
m
m pm akpm
x
1
2 121
2 122 1
2( )
cos lnπ
aaxk
ma
x am a
k
m
cos
ln( )( )
22 1
2 1
2
1
2
π+
+⎛⎝⎜
⎞⎠⎟
+ −+
=∑
mm p− +1
where 0 < p � 2m + 1.
TABLES OF SPECIAL INDEFINITE INTEGRALS
89
(17) Integrals Involving sin ax
17.17.1. sincos
ax dxax
a= −∫
17.17.2. x ax dxax
ax ax
asin
sin cos= −∫ 2
17.17.3. x ax dxx
aax
axa
ax22 3
22 2sin sin cos= + −⎛
⎝⎜⎞⎠⎟∫
17.17.4. x ax dxx
a aax
xa
xa
32
2 4 3
33 6 6sin sin= −⎛
⎝⎜⎞⎠⎟
+ −⎛⎝⎜
⎞⎠⎠⎟∫ cos ax
17.17.5. sin ( )
!( )
!ax
xdx ax
ax ax= −⋅
+⋅
− ⋅ ⋅ ⋅∫3 5
3 3 5 5
17.17.6. sin sin cos
(ax
xdx
axx
aax
xdx
2= − + ∫∫ See 17.18.5.)
17.17.7.dx
axax ax
axsin
ln(csc cot ) ln tan= − =∫1 1
2α α
17.17.8. x dx
ax aax
ax ax n
sin( ) ( ) (
= + + + +−1
1871800
2 22
3 5 2 1
�−−
++
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
+
∫1
2 1
2 1) ( )
( )!
B ax
nn
n
�
17.17.9. sinsin2
22
4ax dx
x axa
= −∫
17.17.10. x ax dxx x ax
aax
asin
sin cos22
242
42
8= − −∫
17.17.11. sincos cos3
3
3ax dx
axa
axa∫ = − +
17.17.12. sinsin sin4 3
82
44
32∫ = − +ax dxx ax
aaxa
17.17.13.dx
ax aax
sincot2
1∫ = −
17.17.14.dx
axax
a ax aax
sincossin
ln tan3 221
2 2= − +∫
17.17.15. sin sinsin ( )
( )sin ( )
(px qx dx
p q xp q
p q xp q
= −−
− ++2 2 ))
( ,∫ = ±If see 17.17.9.)p q
17.17.16.dx
ax aax
11
4 2− = +⎛⎝⎜
⎞⎠⎟∫ sin
tanπ
17.17.17.xdx
axxa
axa
ax1 4 2
24 22−
= +⎛⎝⎜
⎞⎠⎟
+ −sin
tan ln sinπ π⎛⎛
⎝⎜⎞⎠⎟∫
17.17.18.dx
axax
11
4 2+ = − −⎛⎝⎜
⎞⎠⎟∫ sin
tanαπ
17.17.19.x dx
axxa
axa
ax1 4 2
242+ = − −⎛
⎝⎜⎞⎠⎟ + +
sintan ln sin
π π22
⎛⎝⎜
⎞⎠⎟∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
90
17.17.20.dx
ax aax
a( sin )tan tan
11
2 4 21
6 423
− = +⎛⎝⎜
⎞⎠⎟ + +∫
π π aax2
⎛⎝⎜
⎞⎠⎟
17.17.21.dx
ax aax
a( sin )tan tan
11
2 4 21
6 423
+= − −
⎛⎝⎜
⎞⎠⎟
−∫π π −−
⎛⎝⎜
⎞⎠⎟
ax2
17.17.22.dx
p q ax
a p q
p ax q
p q
a q
+ =−
+−
∫
−
sin
tantan2
1
2 2
112
2 2
2 −−+ − −+ + −
⎛
⎝⎜
⎞
⎠p
p ax q q p
p ax q q p2
12
2 2
12
2 2ln
tan
tan ⎟⎟
⎧
⎨
⎪⎪
⎩
⎪⎪
(If p = ± q, see 17.17.16 and 17.17.18.)
17.17.23.dx
p q axq ax
a p q p q axp
p( sin )cos
( )( sin )+ = − + + −2 2 2 2 qqdx
p q ax2 +∫∫ sin
(If p = ± q, see 17.17.20 and 17.17.21.)
17.17.24.dx
p q ax ap p q
p q axp2 2 2 2 2
12 21
+ =+
+−∫ sintan
tan
17.17.25.dx
p q ax
ap p q
p q axp
ap q
2 2 2
2 2
12 21
1
2
− =−
−−
sin
tantan
22 2
2 2
2 2−− +− −
⎛
⎝⎜
⎞
⎠⎟
⎧
⎨
⎪⎪
⎩
⎪p
q p ax p
q p ax pln
tan
tan⎪⎪
∫
17.17.26. x ax dxx ax
amx ax
am m
axm
m m
sincos sin ( )= − + − −∫
−1
2 2
1 mm ax dx−∫ 2 sin
17.17.27.sin sin
( )cosax
xdx
axn x
an
axx
dxn n n∫ ∫= −
−+
−− −1 11 1((See 17.18.30.)
17.17.28. sinsin cos
sinnn
nax dxax axan
nn
ax dx= − + −−−∫ ∫
121
17.17.29. dxax
axa n ax
nn
dxn n nsin
cos( )sin sin
= −− + −
−− −1211 22 ax∫∫
17.17.30. x dxax
x axa n ax a n nn nsin
cos( )sin ( )(
= −− − − −−1
111 2 22
212 2) sin sinn nax
nn
x dxax− −+ −
−∫ ∫
(18) Integrals Involving cos ax
17.18.1. cossin
ax dxax
a=∫
17.18.2. x ax dxax
ax ax
acos
cos sin= +∫ 2
17.18.3. x ax dxx
aax
xa a
ax22
2
3
2 2cos cos sin= + −⎛
⎝⎜⎞⎠⎟∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
91
17.18.4. x ax dxx
a aax
xa
xa
32
2 4
3
3
3 6 6cos cos= −⎛
⎝⎜⎞⎠⎟
+ −⎛⎝⎜
⎞⎠⎠⎟∫ sin ax
17.18.5. cosln
( )!
( )!
( )!
axx
dx xax ax ax= − ⋅ + ⋅ − ⋅ +
2 4 6
2 2 4 4 6 6��∫
17.18.6. cos cos sin
(ax
xdx
axx
aax
xdx
2= − − ∫∫ See 17.17.5.)
17.18.7. dx
ax aax ax
aax
cosln (sec tan ) ln tan= + = +⎛
⎝⎜⎞1 1
4 2π
⎠⎠⎟∫
17.18.8.x dx
ax aax ax ax E axn
cos( ) ( ) ( ) (
= + + + +12 8
51442
2 4 6
�))
( )( )!
2 2
2 2 2
n
n n
+
+ +⎧⎨⎩
⎫⎬⎭∫ �
17.18.9. cossin2
22
4ax dx
x axa
= +∫
17.18.10. x ax dxx x ax
aax
acos
sin cos22
242
42
8= + +∫
17.18.11. cossin sin3
3
3ax dx
axa
axa
= −∫
17.18.12. cossin sin4 3
82
44
32ax dx
x axa
axa
= + +∫
17.18.13. dxax
axacos
tan2 =∫
17.18.14.dx
axax
a ax aax
cossincos
ln tan3 221
2 4 2= + +⎛
⎝⎜⎞⎠⎟
π∫∫
17.18.15. cos cossin( )
( )sin( )
(ax px dx
a p xa p
a p xa∫ = −
−+ +
+2 2 ppa p
)( ,If see 17.18.9.)= ±
17.18.16.dx
ax aax
11
2− = −∫ coscot
17.18.17.x dx
axxa
axa
ax1 2
222− = − +∫ cos
cot ln sin
17.18.18.dx
ax aax
11
2+ =∫ costan
17.18.19.x dx
axxa
axa
ax1 2
222+ = +∫ cos
tan ln cos
17.18.20.dx
ax aax
aax
( cos )cot cot
11
2 21
6 223
− = − −∫
17.18.21.dx
ax aax
aax
( cos )tan tan
11
2 21
6 223
+ = +∫
17.18.22. dxp q ax
a p qp q p q ax
a
+=
−− +−
cos
tan ( ) / ( ) tan2 1
2
1
2 2
1
qq p
ax q p q p
ax q p q2 2
12
12−
+ + −
− +ln
tan ( ) / ( )
tan ( ) / ( −−
⎛
⎝⎜
⎞
⎠⎟
⎧
⎨
⎪⎪
⎩
⎪⎪
∫p)
( ,If see 17.18.16and 17.18.18.)
p q= ±
TABLES OF SPECIAL INDEFINITE INTEGRALS
92
17.18.23.dx
p q axq ax
a q p p q axp
q( cos )sin
( )( cos )+ = − + − −2 2 2 2 ppdx
p q ax2 +∫∫ cos(If see 17.18.19and 17.18.20.)
p q= ±
17.18.24.dx
p q ax ap p q
p ax
p q2 2 2 2 2
1
2 2
1+ =
+ +−∫ cos
tantan
17.18.25.dx
p q ax
ap p q
p ax
p q
ap q
2 2 2
2 2
1
2 2
1
1
2
− =− −
−
cos
tantan
22 2
2 2
2 2−− −+ −
⎛
⎝⎜
⎞
⎠⎟
⎧
⎨⎪⎪
⎩⎪
p
p ax q p
p ax q pln
tan
tan⎪⎪
∫
17.18.26. x ax dxx ax
amx
aax
m ma
xmm m
m∫ = + − −−
cossin
cos( )1
2 2
1 −−∫ 2 cos ax dx
17.18.27.cos cos
( )sin
(ax
xdx
axn x
an
axx
dxn n n
= −−
−−− −1 11 1
Seee 17.17.27.)∫∫
17.18.28. cossin cos
cosnn
nax dxax ax
ann
nax dx= + −−
−∫ ∫1
21
17.18.29.dx
axax
a n axnb
dxn n ncos
sin( ) cos cos
= − + −−− −1
211 2 aax∫∫
17.18.30.x dx
axx ax
a n ax a n nn ncossin
( ) cos ( )(= − − − −−1
11 21 2 )) cos cosn nax
nn
x dxax− −+ −
−∫ ∫2 2
21
(19) Integrals Involving sin ax and cos ax
17.19.1. sin cossin
ax ax dxax
a=∫
2
2
17.19.2. sin coscos( )
( )cos( )
(px qx dx
p q xp q
p q xp
= − −− − +
+2 2 qq)
17.19.3. sin cossin( )
( ,nn
ax ax dxax
n an∫ =
+= −
+1
11If see 117.21.1.)
17.19.4. cos sincos( )
,nn
ax ax dxax
n an∫ = −
+= −
+1
11(If see 17.20.1.)
17.19.5. sin cossin2 2
84
32ax ax dx
x axa
= −∫
17.19.6.dx
ax ax aax
sin cosln tan=∫
1
17.19.7.dx
ax ax aax
a axsin cosln tan
sin2
14 2
1= +⎛⎝⎜
⎞⎠⎟ −∫
π
17.19.8.dx
ax ax aax
a axsin cosln tan
cos2
12
1= +∫
17.19.9.dx
ax axax
asin coscot
2 2
2 2= −∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
93
17.19.10.sincos
sinln tan
2 12 4
axax
dxax
a aax= − + +⎛
⎝⎜⎞⎠⎟∫
π
17.19.11.cossin
cosln tan
2 12
axax
dxax
a aax= +∫
17.19.12.dx
ax ax a ax aax
cos ( sin ) ( sin )ln tan
11
2 11
2 2± = ± +∫ ∓ ++⎛⎝⎜
⎞⎠⎟
π4
17.19.13.dx
ax ax a ax aax
sin ( cos ) ( cos )ln tan
11
2 11
2 2± = ± ± +∫
17.19.14.dx
ax ax a
axsin cos
ln tan± = ±⎛⎝⎜
⎞⎠⎟∫
1
2 2 8π
17.19.15.sin
sin cosln (sin cos )
ax dxax ax
xa
ax ax± = ±∫ 21
2∓
17.19.16.cos
sin cosln (sin cos )
ax dxax ax
xa
ax ax± = ± + ±∫ 21
2
17.19.17.sin
cosln ( cos )
ax dxp q ax aq
p q ax+ = − +∫1
17.19.18.cos
sinln ( sin )
ax dxp q ax aq
p q ax+ = +∫1
17.19.19.sin
( cos ) ( )( cos )ax dx
p q ax aq n p q axn n+ = − + −∫1
1 1
17.19.20.cos
( sin ) ( )( sin )ax dx
p q ax aq n p q axn n+ = −− + −∫
11 1
17.19.21.dx
p ax q ax a p q
ax q psin cos
ln tantan ( / )
+=
++⎛ −1
22 2
1
⎝⎝⎜⎞⎠⎟∫
17.19.22.dx
p ax q ax r
a r p q
p r q
sin cos
tan( ) tan
+ +=
− −+ −−2
2 2 2
1 (( / )
ln
ax
r p q
a p q r
p p q r
2
1
2 2 2
2 2 2
2 2 2
− −
⎛
⎝⎜
⎞
⎠⎟
+ −
− + − ++ −
+ + − + −
⎛
⎝
( ) tan ( / )
( ) tan ( / )
r q ax
p p q r r q ax
2
22 2 2⎜⎜
⎞
⎠⎟
⎧
⎨
⎪⎪
⎩
⎪⎪
∫
(If r = q see 17.19.23. If r2 = p2 + q2 see 17.19.24.)
17.19.23.dx
p ax q ax apq p
axsin ( cos )
ln tan+ + = +⎛⎝⎜
⎞⎠⎟∫ 1
12
17.19.24.dx
p ax q ax p q a p q
ax
sin costan
tan
+ ± += −
++ −
2 2 2 2
14π
∓11
2( / )q p⎛
⎝⎜⎞⎠⎟∫
17.19.25.dx
p ax q ax apqp ax
q2 2 2 211
sin costan
tan+ = ⎛
⎝⎜⎞⎠⎟
−∫
17.19.26.dx
p ax q ax apqp ax qp ax q2 2 2 2
12sin cos
lntantan− = −
+⎛⎛⎝⎜
⎞⎠⎟∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
94
17.19.27. sin cos
sin cos( )m n
m n
ax ax dx
ax axa m n
mm=
− + + −− +1 1 1++
+ +
−
+ −
∫nax ax dx
ax axa m n
m n
m n
sin cos
sin cos( )
2
1 1 nnm n
ax ax dxm n−+
⎧
⎨⎪
⎩⎪ −∫
∫ 1 2sin cos
17.19.28.sincos
sin( ) cos
m
n
m
n
axax
dx
axa n ax
mn
=
− − −−
−
−
1
11111
1
2
2
1
1
sincos
sin( ) cos
m
n
m
n
axax
dx
axa n ax
−
−
+
−
∫
− −− − +−
−−
−
−
∫m n
naxax
dx
axa m n
m
n
m
21 2
1
sincos
sin( ) coos
sincosn
m
naxmm n
axax
dx−
−
+ −−
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪ ∫
∫
1
21
17.19.29.cossin
cos( )sin
m
n
m
n
axax
dx
axa n ax
mn
=
−− − −−
−
1
111
−−−
−
−
−
+
−
∫1
1
2
2
1
1
cossin
cos( )sin
m
n
m
n
axax
dx
axa n aax
m nn
axax
dx
axa m n
m
n
m
− − +−
−
−
−
∫2
1 2
1
cossin
cos( )ssin
cossinn
m
naxmm n
axax
dx−
−
+ −−
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪ ∫
∫
1
21
17.19.30.dx
ax axa n ax ax
m n
m n
m n
sin cos( )sin cos= − + + −
− −1
1 1 1
221
11
2
1
ndx
ax ax
a m ax
m n
m n
−−
−
−
− −
∫ sin cos
( )sin cos 11 2
21ax
m nm
dxax axm n+ + −
−
⎧
⎨⎪
⎩⎪⎪ −∫
∫sin cos
(20) Integrals Involving tan ax
17.20.1. tan ln cos ln secax dxa
axa
ax= − =∫1 1
17.20.2. tantan2 ax dx
axa
x= −∫
17.20.3. tantan
ln cos32
21
ax dxax
a aax= +∫
17.20.4. tan sectan( )
nn
ax ax dxax
n a2
1
1= +
+
∫
17.20.5.sectan
ln tan2 1axax
dxa
ax=∫
17.20.6.dx
ax aax
tanln sin=∫
1
17.20.7. x ax dxa
ax ax ax n
tan( ) ( ) ( ) (
= + + + +13 15
2105
22
3 5 7 2
�22 1
2 1
2 2 1nn
nB axn
−+ +⎧
⎨⎩
⎫⎬⎭
+
∫) ( )
( )!�
17.20.8.tan ( ) ( ) ( ) (ax
xdx ax
ax ax Bn nn= + + + +
−3 5 2 2
92
752 2 1
�aax
n n
n)( )( )!
2 1
2 1 2
−
− +∫ �
17.20.9. x ax dxx ax
a aax
xtan
tanln cos2
2
212
= + −∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
95
17.20.10.dx
p q axpx
p qq
a p qq ax p a+ = + + + +
tan ( )ln ( sin cos2 2 2 2 xx)∫
17.20.11. tantan( )
tannn
nax dxax
n aax dx=
−−
−−∫∫
12
1
(21) Integrals Involving cot ax
17.21.1. cot ln sinax dxa
ax∫ = 1
17.21.2. cotcot2 ax dx
axa
x= − −∫
17.21.3. cotcot
ln sin32
21
∫ = − −ax dxax
a aax
17.21.4. cot csccot( )
nn
ax ax dxax
n a∫ = −+
+2
1
1
17.21.5. csccot
ln cot2 1axax
dxa
ax= −∫
17.21.6. dx
ax aax
cotln cos= −∫
1
17.21.7. x ax dxa
axax ax B axn
ncot( ) ( ) ( )
∫ = − − − −19 225
22
3 5 2
�22 1
2 1
n
n
+
+−
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪( )!�
17.21.8. cot ( ) ( )
(ax
xdx
axax ax B axn
nn
= − − − − −−1
3 135
23 2 2 1
�22 1 2n n−
−∫ )( )!�
17.21.9. x ax dxx ax
a aax
xcot
cotln sin2
2
212∫ = − + −
17.21.10.dx
p q axpx
p qq
a p qq ax q a
+=
+−
++
cot ( )ln ( sin cos
2 2 2 2xx)∫
17.21.11. cotcot( )
cotnn
nax dxax
n aax dx∫ ∫= −
−−
−−
12
1
(22) Integrals Involving sec ax
17.22.1. sec ln (sec tan ) ln tanax dxa
ax axa
ax= + = +⎛⎝⎜
⎞1 12 4
π⎠⎠⎟∫
17.22.2. sectan2∫ =ax dx
axa
17.22.3. secsec tan
ln (sec tan )3
21
2∫ = + +ax dxax ax
a aax ax
TABLES OF SPECIAL INDEFINITE INTEGRALS
96
17.22.4. sec tansecn
n
ax ax dxax
na∫ =
17.22.5. dx
axax
asecsin=∫
17.22.6. x ax dxa
ax ax ax E axnsec
( ) ( ) ( ) (= + + + +1
2 851442
2 4 6
�))
( )( )!
2 2
2 2 2
n
n n
+
++
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪∫ �
17.22.7.sec
ln( ) ( ) ( )ax
xdx x
ax ax ax= + + + + +2 4 6
45
9661
4320�
EE ax
n nn
n( )
( )!
2
2 2+∫ �
17.22.8. x ax dxxa
axa
axsec tan ln cos22
1∫ = +
17.22.9.dx
q p axxq
pq
dxp q ax+
= −+∫ ∫sec cos
17.22.10. secsec tan
( )secn
nnax dx
ax axa n
nn∫ ∫=
−+ −
−
−−
22
121
aax dx
(23) Integrals Involving csc ax
17.23.1. csc ln (csc cot ) ln tanax dxa
ax axa
ax= − =∫1 1
2
17.23.2. csccot2∫ = −ax dx
axa
17.23.3. csccsc cot
ln tan3
21
2 2∫ = − +ax dxax ax
a aax
17.23.4. csc cotcscn
n
ax ax dxax
na∫ = −
17.23.5. dx
axax
acsccos= −∫
17.23.6. x ax dxa
axax ax n
csc( ) ( ) (
= + + + +∫−1
1871800
2 22
3 5 2
�11 2 11
2 1
−+
+⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
+) ( )
( )!
B ax
nn
n
�
17.23.7. csc ( ) ( )ax
xdx
axax ax Bn
n= − + + + +−−1
671080
2 2 13 2 1
�(( )
( )( )!
ax
n n
n2 1
2 1 2
−
−+∫ �
17.23.8. x ax dxx ax
a aaxcsc
cotln sin2
2
1∫ = − +
17.23.9. dx
q p axxq
pq
dxp q ax+
= −+∫ ∫csc sin
(See 17.17.22.)
17.23.10. csccsc cot
( )cscn
nnax dx
ax axa n
nn
= −−
+ −−
−−∫
22
121 ∫∫ ax dx
TABLES OF SPECIAL INDEFINITE INTEGRALS
97
(24) Integrals Involving Inverse Trigonometric Functions
17.24.1. sin sin− −= + −∫ 1 1 2 2xa
dx xxa
a x
17.24.2. xxa
dxx a x
ax a x
sin sin− −= −⎛⎝⎜
⎞⎠⎟
+ −∫ 12 2
12 2
2 4 4
17.24.3. xxa
dxx x
ax a a x2 1
31
2 2 2 2
32
9∫ − −= + + −sin sin
( )
17.24.4.sin ( / ) ( / ) ( / )−
= + +1 3 5
2 3 31 32 4 5
x ax
dxxa
x a x ai i
ii i ii
i ii i i i
�5
1 3 52 4 6 7 7
7
+ +∫( / )x a
17.24.5.sin ( / ) sin ( / )
ln− −
= − − + −⎛
⎝∫1
2
1 2 21x ax
dxx a
x aa a x
x⎜⎜⎞
⎠⎟
17.24.6. sin sin− −⎛⎝⎜
⎞⎠⎟ = ⎛
⎝⎜⎞⎠⎟ − + −∫ 1
2
1
2
2 22 2xa
dx xxa
x a x ssin−1 xa
17.24.7. cos cos− −= − −∫ 1 1 2 2xa
dx xxa
a x
17.24.8. xxa
dxx a x
ax a x
cos cos− −∫ = −⎛⎝⎜
⎞⎠⎟
− −12 2
12 2
2 4 4
17.24.9. xxa
dxx x
ax a a x2 1
31
2 2 2 2
32
9∫ − −= − + −cos cos
( )
17.24.10.cos ( / )
lnsin ( / )
(− −
= − ∫∫1 1
2x a
xdx x
x ax
dxπ
See 17.224.4.)
17.24.11.cos ( / ) cos ( / )
ln− −
= − + + −⎛
⎝∫1
2
1 2 21x ax
dxx a
x aa a x
x⎜⎜⎞
⎠⎟
17.24.12. cos cos− −⎛⎝⎜
⎞⎠⎟ = ⎛
⎝⎜⎞⎠⎟ − − −∫ 1
2
1
2
2 22 2xa
dx xxa
x a x ccos−1 xa
17.24.13. tan tan ln ( )− −= − +∫ 1 1 2 2
2xa
dx xxa
ax a
17.24.14. xxa
dx x axa
axtan ( ) tan− −= + −∫ 1 2 2 11
2 2
17.24.15. xxa
dxx x
aax a
x a2 13
12 3
2 2
3 6 6tan tan ln ( )− −= − + +∫
17.24.16. tan ( / ) ( / ) ( / ) ( / )−
= − + −1 3
2
5
2
7
23 5 7x a
xdx
xa
x a x a x a ++∫ �
17.24.17.tan ( / )
tan ln−
−= − − +⎛⎝⎜
⎞⎠
1
21
2 2
2
1 12
x ax
dxx
xa a
x ax ⎟⎟∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
98
17.24.18. cot cot )− −∫ = + +1 1 2 2
2xa
dx xxa
ax aln (
17.24.19. xxa
dx x axa
axcot ( ) cot− −∫ = + +1 2 2 11
2 2
17.24.20. xxa
dxx x
aax a
x a2 13
12 3
2 2
3 6 6cot cot ( )− −∫ = + − +ln
17.24.21.cot ( / ) tan ( / )− −
∫ ∫= −1 1
2x a
xdx x
x ax
dxπ
ln (See 17.24.16.)
17.24.22.cot ( / ) cot ( / )− −
∫ = + +⎛⎝⎜
1
2
1 2 2
2
12
x ax
dxx a
x ax a
xln
⎞⎞⎠⎟
17.24.23. secsec ( ) sec
−
− −
=− + − < <
1
1 2 2 102x
adx
xxa
a x x axa
lnπ
xxxa
a x x axa
sec sec− −+ + − < <
⎧
⎨⎪
⎩⎪
∫1 2 2 1
2ln ( )
π π
17.24.24. xxa
dx
x xa
a x a xa
xsec
sec sec−
− −
=− − < <
1
21
2 21
2
2 20
2π
22 2 21
2 21sec sec− −+ − < <
⎧
⎨⎪⎪
⎩⎪⎪
∫xa
a x a xa
π π
17.24.25. xxa
dx
x xa
ax x a ax x
2 1
31
2 2 32
3 6 6secsec
−
−
=− − − + −ln( aa
xa
x xa
ax x a ax
2 1
31
2 2 3
02
3 6 6
) sec
sec
< <
+ − +
−
−
π
ln( ++ − < <
⎧
⎨⎪⎪
⎩⎪⎪
−∫
x axa
2 2 1
2) sec
π π
17.24.26.sec ( / ) ( / ) ( /−
= + + +∫1 3
2 2 3 31 3x a
xdx x
ax
a x a xπln
i ii )) ( / )5 7
2 4 5 51 3 52 4 6 7 7i i ii ii i i i
+ + ⋅⋅ ⋅a x
17.24.27.sec ( / )
sec ( / )sec
−
−−
=
− + − <1
2
1 2 210
x ax
dx
x ax
x aax
xaa
x ax
x aax
xa
<
− − − < <
⎧
⎨
⎪⎪⎪
⎩
−−
π
π π
2
2
1 2 21sec ( / )
sec⎪⎪⎪⎪
∫
17.24.28. csccsc ( ) csc
−
− −
=+ + − < <
1
1 2 2 102x
adx
xxa
a x x axa
lnπ
xxxa
a x x axa
csc ( ) csc− −− + − − < <
⎧
⎨⎪
⎩⎪
∫1 2 2 1
20ln
π
17.24.29. xxa
dx
x xa
a x a xa
xcsc
csc csc−
− −
=+ − < <
1
21
2 21
2
2 20
2π
22 2 201
2 21csc csc− −− − − < <
⎧
⎨⎪⎪
⎩⎪⎪
∫xa
a x a xa
π
17.24.30. xxa
dx
x xa
ax x a ax x
2 1
31
2 2 32
3 6 6csc
csc (−
−
=+ − + +
∫ln −− < <
− − −
−
−
axa
x xa
ax x a a
2 1
31
2 2 3
02
3 6 6
) csc
csc (
π
ln xx x axa
+ − − < <
⎧
⎨⎪⎪
⎩⎪⎪ −2 2 1
20) csc
π
TABLES OF SPECIAL INDEFINITE INTEGRALS
99
17.24.31.csc ( ) ( ) ( )−
∫ = − + +1 3 51 3
4x a
xdx
ax
a x a x/ /2 3 3
/2i iii ii i
i ii i i i5 5
/2 6 7 7
+ + ⋅⋅ ⋅⎛⎝⎜
⎞⎠⎟
1 3 54
7( )a x
17.24.32.csc ( )
csc ( )csc
−
−−
∫ =− − − <
1
2
1 2 210
x ax
dx
x ax
x aax/
/ xxa
x ax
x aax
xa
<
− + − − < <
⎧
⎨⎪⎪
−−
π
π
2
20
1 2 21csc ( )
csc/
⎩⎩⎪⎪
17.24.33. xxa
dxxm
xa m
x
a xdxm
m m
sin sin−+
−+
= + − + −∫11
11
2 211
1∫∫
17.24.34. xxa
dxxm
xa m
x
a xdxm
m m
cos cos−+
−+
=+
++ −∫ 1
11
1
2 211
1 ∫∫
17.24.35. xxa
dxxm
xa
am
xx a
dxmm m
tan tan−+
−+
=+
−+ +∫ 1
11
1
2 21 1 ∫∫
17.24.36. xxa
dxxm
xa
am
xx a
dxmm m
cot cot−+
−+
=+
++ +∫ 1
11
1
2 21 1 ∫∫
17.24.37. xxa
dx
x x am
am
x dx
x am
m m
sec
sec ( / )
−
+ −
=+
−+ −
∫ 1
1 1
21 1 22
1
1 1
2
02
1 1
∫ < <
++
+
−
+ −
sec
sec ( / )
xa
x x am
am
x dx
x
m m
π
−−< <
⎧
⎨⎪⎪
⎩⎪⎪ ∫ −
a
xa2
1
2π πsec
17.24.38. xxa
dx
x x am
am
x dx
x am
m m
csc
sec ( )
−
+ −
=+
++ −
∫ 1
1 1
21 1/
22
1
1 1
2
02
1 1
∫ < <
+−
+
−
+ −
csc
csc ( )
xa
x x am
am
x dx
x
m m
π
/
−−− < <
⎧
⎨⎪⎪
⎩⎪⎪ ∫ −
a
xa2
1
20
πcsc
(25) Integrals Involving eax
17.25.1. e dxea
axax
∫ =
17.25.2. xe dxea
xa
axax
∫ = −⎛⎝⎜
⎞⎠⎟
1
17.25.3. x e dxea
xx
a aax
ax2 2
2
2 2∫ = − +
⎛⎝⎜
⎞⎠⎟
17.25.4. x e dxx e
ana
x e dx
ea
xnx
a
n axn ax
n ax
axn
n
∫ ∫= −
= − +
−
−
1
1 nn n xa
na
nn n
n
( ) ( ) !− − ⋅⋅ ⋅ −⎛⎝⎜
⎞⎠⎟
=−1 12
2 if positivee integer
17.25.5.ex
dx xax ax axax
= + + + + ⋅⋅ ⋅∫ ln1 1 2 2 3 3
2 3
i i i!( )
!( )
!
17.25.6.ex
dxe
n xa
nex
dxax
n
ax
n
ax
n∫ ∫= −− + −− −( )1 11 1
TABLES OF SPECIAL INDEFINITE INTEGRALS
100
17.25.7. dx
p qexp ap
p qeax
ax
+= − +∫
1ln ( )
17.25.8. dx
p qexp ap p qe ap
p qeax ax
ax
( ) ( ))
+= +
+− +∫ 2 2 2
1 1ln (
17.25.9.dx
pe qe
a pq
pq
e
a pq
eax ax
ax
+ =
⎛⎝⎜
⎞⎠⎟
−
−
−
∫
1
1
2
1tan
lnaax
ax
q p
e q p
− −+ −
⎛
⎝⎜⎞
⎠⎟
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
/
/
17.25.10. e bx dxe a bx b bx
a bax
ax
sin( sin cos )= −
+∫ 2 2
17.25.11. e bx dxe a bx b bx
a bax
ax
cos( cos sin )= +
+∫ 2 2
17.25.12. xe bx dxxe a bx b bx
a beax
ax ax
sin( sin cos ) {(= −
+−∫ 2 2
aa b bx ab bxa b
2 2
2 2 2
2− −+
) sin cos }( )
17.25.13. xe bx dxxe a bx b bx
a beax
ax ax
cos( cos sin ) {(= +
+ −∫ 2 2
aa b bx ab bxa b
2 2
2 2 2
2− ++
) cos sin }( )
17.25.14. e x dxe x
a aex
dxaxax ax
lnln∫ ∫= − 1
17.25.15. e bx dxe bxa n b
a bx nbax nax n
∫ = + −−
sinsin
( sin co1
2 2 2 ss )( )
sinbxn n ba n b
e bx dxax n+ −+
−∫1 2
2 2 22
17.25.16. e bx dxe bx
a n ba bx nbax n
ax n
∫ = + +−
coscos
( cos si1
2 2 2 nn )( )
cosbxn n ba n b
e bx dxax n+ −+
−∫1 2
2 2 22
(26) Integrals Involving ln x
17.26.1. ln lnx dx x x x= −∫
17.26.2. x x dxx
xln ln= −⎛⎝⎜
⎞⎠⎟∫
2
212
17.26.3. x x dxxm
xm
mmm
ln ln If see=+
−+
⎛⎝⎜
⎞⎠⎟
= −+
∫1
11
11 17( , .. . .)26 4
17.26.4. lnln
xx
dx x=∫12
2
17.26.5.ln lnxx
dxx
x x2
1= − −∫17.26.6. ln ln ln2 2 2 2x dx x x x x x∫ = − +
17.26.7.ln ln
If seen nx dxx
xn
n=+
= −+
∫1
11 17 26 8( , . . .)
17.26.8.dx
x xx
lnln ln=∫ ( )
TABLES OF SPECIAL INDEFINITE INTEGRALS
101
17.26.9. dx
xx x
x xln
ln ln= + + + + ⋅⋅ ⋅∫ ( ) lnln
!ln
!
2 3
2 2 3 3i i
17.26.10.x dx
xx m x
m x mm
lnln ln= + + + + + +
( ) ( ) ln( ) ln
!(
11
2 2
2 2
i11
3 3
3 3) ln!
xi∫ + ⋅ ⋅ ⋅
17.26.11. ln lnn n nx dx x x n x dx= − −∫∫ ln 1
17.26.12. x x dxx x
mn
mx xdxm n
m nm nln
lnln=
+−
+
+−∫ ∫
11
1 1
If m = –1, see 17.26.7.
17.26.13. ln ( ln (x a dx x x a x axa
2 2 2 2 12 2+ = + − + −∫ ) ) tan
17.26.14. ln ( ) ln ( ) lnx a dx x x a x ax ax a
2 2 2 2 2− = − − + +−
⎛⎝⎜
⎞⎠⎟∫
17.26.15. x x a dxx x a
m mx
x am
m m
ln )ln ( )
( 2 21 2 2 2
212
1± = ±
+ − + ±+ +
22 dx∫∫
(27) Integrals Involving sinh ax
17.27.1. sinhcosh
ax dxax
a∫ =
17.27.2. x ax dxx ax
aax
asinh
cosh sinh∫ = −2
17.27.3. x ax dxxa a
axx
aax2
2
3 2
2 2sinh cosh sinh∫ = +
⎛⎝⎜
⎞⎠⎟
−
17.27.4.sinh ( )
!( )
!ax
xdx ax
ax ax= + + + ⋅⋅ ⋅∫3 5
3 3 5 5i i
17.27.5.sinh sinh coshax
xdx
axx
aax
xdx2 = − +∫ ∫
(See 17.28.4.)
17.27.6.dx
ax aax
sinhln tanh=∫
12
17.27.7.x dx
ax aax
ax axsinh
( ) ( ) (= − + − ⋅⋅ ⋅ +
−∫
118
71800
22
3 5 11 2 12 1
2 2 1) ( ) ( )( )!
n nn
nB axn−
+ + ⋅⋅ ⋅⎧⎨⎩
⎫⎬⎭
+
17.27.8. sinhsinh cos2
2 2ax dx
ax axa
x= −∫h
17.27.9. x ax dxx ax
aax
ax
sinhsinh cosh2
2
224
28 4
= − −∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
102
17.27.10.dx
axax
asinhcoth
2= −∫
17.27.11. sinh sinhsinh ( )
( )sinh ( )
ax px dxa p x
a pa p x= +
+− −
∫ 2 22( )a p−
For a = ± p see 17.27.8.
17.27.12. x ax dxx ax
ama
x ax dxmm
msinhcosh
cosh= −∫ ∫ −1 (See 17.28.12.)
17.27.13. sinhsinh cosh
sinhnn
nax dxax axan
nn
ax d= − −∫
−−
121
xx∫
17.27.14. sinh sinh( )
coshaxx
dxax
n xa
nax
xn n n∫ = −−
+−− −1 11 1
ddx∫ (See 17.28.14.)
17.27.15.dx
axax
a n axnn
dxn nsinh
cosh( )sinh s∫ = −
−− −
−−1211 iinhn ax−∫ 2
17.27.16.x dx
axx ax
a n ax a nn nsinhcosh
( )sinh (∫ = −−
−−−1
111 2 ))( )sinh sinhn ax
nn
x dxaxn n−
− −−− −∫2
212 2
(28) Integrals Involving cosh ax
17.28.1. coshsinh
ax dxax
a∫ =
17.28.2. x ax dxx ax
aax
acosh
sinh cosh∫ = −2
17.28.3. x ax dxx ax
axa a
ax22
2
3
2 2cosh
coshsinh∫ = − + +
⎛⎝⎜
⎞⎠⎟
17.28.4.cosh
ln( )
!( )
!( )ax
xdx x
ax ax ax= + + +∫2 4 6
2 2 4 4 6 6i i i !!+ ⋅⋅ ⋅
17.28.5.cosh cosh sinhax
xdx
axx
aax
xdx2 = − +∫ ∫ (See 17.27.4.)
17.28.6.dx
ax aeax
coshtan= −∫
2 1
17.28.7.x dx
ax aax ax ax
cosh( ) ( ) ( ) (
= − + + ⋅⋅ ⋅ +∫1
2 851442
2 4 6 −−+ + ⋅⋅ ⋅⎧
⎨⎩
⎫⎬⎭
+12 2 2
2 2) ( )( )( )!
nn
nE axn n
17.28.8. coshsinh cosh2
2 2ax dx
x ax axa
= +∫
17.28.9. x ax dxx x ax
aax
acosh
sinh cosh22
242
42
8= + −∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
103
17.28.10.dx
axax
acoshtanh
2=∫
17.28.11. cosh coshsinh( )
( )sinh( )
ax px dxa p xa p
a p x= −−
+ +∫ 2 22( )a p+
17.28.12. x ax dxx ax
ama
x ax dxmm
mcoshsinh
sinh= −∫ ∫ −1 (See 17.27.12.)
17.28.13. coshcosh sinh
coshnn
nax dxax axan
nn
ax d= + −∫−
−1
21xx∫
17.28.14.cosh cosh
( )sinhax
xdx
axn x
an
axxn n n∫ = −
−+
−− −1 11 1ddx∫ (See 17.27.14.)
17.28.15.dx
axax
a n axnn
dxn ncosh
sinh( ) cosh co∫ =
−+ −
−−1211 sshn ax−∫ 2
17.28.16.x dx
axx ax
a n ax n nn ncoshsinh
( ) cosh ( ) (∫ =−
+−−1
111 −−
+ −−− −∫2
212 2 2) cosh cosha ax
nn
x dxaxn n
(29) Integrals Involving sinh ax and cosh ax
17.29.1. sinh coshsinh
ax ax dxax
a=∫
2
2
17.29.2. sinh coshcosh ( )
( )cosh ( )
px qx dxp q x
p qp q x= +
++ −
∫ 2 22( )p q−
17.29.3. sinh coshsinh2 2 4
32 8ax ax dx
axa
x= −∫
17.29.4.dx
ax ax aax
sinh coshln tanh∫ = 1
17.29.5.dx
ax axax
asinh coshcoth
2 2
2 2∫ = −
17.29.6. sinhcosh
sinhtan sinh
211ax
axdx
axa a
ax∫ = −
17.29.7.coshsinh
coshln tanh
2 12
axax
dxax
a aax∫ = +
TABLES OF SPECIAL INDEFINITE INTEGRALS
104
(30) Integrals Involving tanh ax
17.30.1. tanh ln coshax dxa
ax=∫1
17.30.2. tanhtanh2 ax dx x
axa
= −∫
17.30.3. tanh ln coshtanh3
212
ax dxa
axax
a= −∫
17.30.4. x ax dxa
ax ax axtanh
( ) ( ) ( ) (= − + − ⋅⋅ ⋅
−13 15
21052
3 5 7 11 2 2 1
2 1
1 2 2 2 1) ( ) ( )
( )!
n n nn
nB ax
n
− +−+
+ ⋅ ⋅ ⋅⎧⎨⎪
⎩⎪
⎫⎬⎬⎪
⎭⎪∫
17.30.5. x ax dxx x ax
a aaxtanh
tanhln cosh2
2
221= − +∫
17.30.6.tanh ( ) ( ) ( ) (ax
xdx ax
ax ax n n
= − + − ⋅⋅ ⋅− −3 5 1 2
92
75
1 2 22 1
2 1 2
2 2 1nn
nB ax
n n
−−
+ ⋅⋅ ⋅−
∫) ( )
( )( )!
17.30.7.dx
p q axpx
p qq
a p qq ax p+ = − − − +∫ tanh ( )
ln ( sinh c2 2 2 2 oosh )ax
17.30.8. tanhtanh
( ) tanhnn
nax dxax
a aax dx= −
− +−
−∫∫1
2
1
(31) Integrals Involving coth ax
17.31.1. coth ln sinhax dxa
ax∫ = 1
17.31.2. cothcoth2 ax dx x
axa∫ = −
17.31.3. coth ln sinhcoth3
212
ax dxa
axax
a∫ = −
17.31.4. x ax dxa
axax ax n
coth( ) ( ) ( )
∫ = + − + ⋅⋅ ⋅− −1
9 2251
2
3 5 1222 1
2 2 1nn
nB axn
( )( )!
+
+ + ⋅ ⋅ ⋅⎧⎨⎩
⎫⎬⎭
17.31.5. x ax dxx x ax
a aaxcoth
cothln sinh2
2
221∫ = − +
17.31.6.coth ( ) ( ) (ax
xdx
axax ax B an n
n= − + − + ⋅⋅ ⋅−1
3 1351 23 2 xx
n n
n)( )( )!
2 1
2 1 2
−
− + ⋅ ⋅ ⋅∫
17.31.7.dx
p q axpx
p qq
a p qp ax q+ = − − − +∫ coth ( )
ln ( sinh c2 2 2 2 oosh )ax
17.31.8. cothcoth
( )cothn
nnax dx
axa n
ax dx= − − +−
−∫∫1
2
1
TABLES OF SPECIAL INDEFINITE INTEGRALS
105
(32) Integrals Involving sech ax
17.32.1. sech ax dxa
eax=∫ −2 1tan
17.32.2. sech2 ax dxax
a=∫
tanh
17.32.3. sech3 ax dxax ax
a aax= +∫ −sech tanh
tan sinh2
12
1
17.32.4. x ax dxa
ax ax axsech∫ = − + + ⋅⋅ ⋅
−12 8
51442
2 4 6( ) ( ) ( ) ( 112 2 2
2 2) ( )( )( )!
nn
nE axn n
+
+ + ⋅ ⋅ ⋅⎧⎨⎩
⎫⎬⎭
17.32.5. x ax dxx ax
a aaxsech2 = −∫
tanhln cosh
12
17.32.6.sech
ln( ) ( ) ( )ax
xdx x
ax ax ax= − + − + ⋅2 4 6
45
9661
4320⋅⋅ ⋅
−+ ⋅ ⋅ ⋅∫
( ) ( )( )!
12 2
2nn
nE axn n
17.32.7. sech sechnn
ax dxax ax
a nnn
= − + −−∫
−sech tanh( )
2
121
nn ax dx−∫ 2
(33) Integrals Involving csch ax
17.33.1. csch ax dxa
ax=∫1
2ln tanh
17.33.2. csch2 ax dxax
a= −∫
coth
17.33.3. csch3 ax dxax ax
a aax= − −∫
csch cothln tanh
21
2 2
17.33.4. x ax dxa
axax ax
csch = − + + ⋅⋅ ⋅ +−
∫1
1871800
22
3 5( ) ( ) ( 11 2 12 1
2 1 2 1) ( ) ( )( )!
n nn
nB axn
− +−+ + ⋅ ⋅ ⋅⎧
⎨⎩
⎫⎬⎭
17.33.5. x ax dxx ax
a aaxcsch2 = − +∫
cothlnsinh
12
17.33.6.csch ( ) ( ) (ax
xdx
axax ax n n
= − − + + ⋅⋅ ⋅−1
671080
1 2 23 2 −− −−− + ⋅ ⋅ ⋅∫1 2 11
2 1 2) ( )
( ) ( )!B ax
n nn
n
17.33.7. csch cscnn
ax dxax ax
a nnn
= −− − −
−∫−csch coth( )
2
121
hh 2n ax dx−∫
TABLES OF SPECIAL INDEFINITE INTEGRALS
106
(34) Integrals Involving Inverse Hyperbolic Functions
17.34.1. sinh sinh− −∫ = − +1 1 2 2xa
dx xxa
x a
17.34.2. xxa
dxx a x
ax
x asinh sinh− −∫ = +⎛
⎝⎜⎞⎠⎟
− +12 2
12 2
2 4 4
17.34.3.sinh ( )
( ) ( )
−
∫ =
− +
1
3 5
2 3 31 32 4
x ax
dx
xa
x a x a
/
/ /i i
ii ii i
i ii i i i5 5
1 3 52 4 6 7 7
2
7
2
− + ⋅⋅ ⋅( )
ln (
x ax a
x a
/| | <
/ )) ( ) ( ) ( )2 2 2 2
1 32 4 4 4
1 3 52 4
− + −a x a x a x/ / /i i
ii i i
i i 66
2 2
2 4 6 6 6
22 2 2 2
i i i i
i i
+ ⋅⋅ ⋅
− − + −
x a
x a a x
>
/ /ln ( ) ( ) 11 32 4 4 4
1 3 52 4 6 6 6
4 6ii i i
i ii i i i
( ) ( )a x a xx
/ /+ − ⋅⋅ ⋅ << −
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
a
17.34.4. coshcosh ( ) , cosh ( )
−
− −
∫ =− − >
1
1 2 2 1xa
dxx x a x a x a/ / 00
01 2 2 1x x a x a x acosh ( ) , cosh ( )− −+ − <
⎧⎨⎪
⎩⎪ / /
17.34.5. xxa
dxx a x a x x a
cosh( )cosh ( )
−
−
∫ =− − −
1
2 2 1 2 214 2
14/ ,, cosh ( )
( )cosh ( )
−
−
>
− + −
1
2 2 1 2
0
14 2
14
x a
x a x a x x
/
/ aa x a2 1 0, cosh ( )− <
⎧
⎨⎪⎪
⎩⎪⎪ /
17.34.6.
cosh ( )ln ( )
( )−
∫ = ± + +1
221
22
1x ax
dx x aa x/
//
2 2 2i ii33 1 3 54 6( ) ( )a x a x/
2 4 4 4/
2 4 6 6 6i i ii ii i i i+ + ⋅⋅ ⋅⎡
⎣⎢⎤⎤⎦⎥
+ > − <− −if / if /cosh ( ) , cosh ( )1 10 0x a x a
17.34.7. tanh tanh ln( )− −∫ = + −1 1 2 2
2xa
dx xxa
aa x
17.34.8. xxa
dxax
x axa
tanh ( ) tanh− −∫ = + −1 2 2 1
212
17.34.9.tanh ( ) ( ) ( )−
∫ = + + + ⋅⋅ ⋅1 3
2
5
23 5x a
xdx
xa
x a x a/ / /
17.34.10. coth coth ln ( )− −∫ = + −1 1 2 2
2xa
dx x xa
x a
17.34.11. xxa
dxax
x axa
coth ( )coth− −∫ = + −1 2 2 1
212
17.34.12.coth ( ) ( ) ( )−
∫ = − + + + ⋅⋅ ⋅⎛⎝
1 3
2
5
23 5x a
xdx
ax
a x a x/ / /⎜⎜
⎞⎠⎟
17.34.13. sechsech ( ) sin ( ), sech
−
− − −
∫ =+
1
1 1 1xa
dxx x a a x a/ / (( )
sech ( ) sin ( ), sech (
x a
x x a a x a x a
/
/ / /
>
−− − −
0
1 1 1 )) <
⎧⎨⎪
⎩⎪ 0
17.34.14. csch sinh ( ,− − −∫ = ± + > −1 1 1 0xa
dx xxa
axa
x xcsch if if << 0)
TABLES OF SPECIAL INDEFINITE INTEGRALS
107
17.34.15. xxa
dxxm
xa m
x
x am
m m
sinh sinh−+
−+
=+
−+ +∫ 1
11
1
2 211
1ddx∫
17.34.16. xxa
dx
xm
xa m
x
x am
m m
cosh
cosh
−
+−
+
∫ =+
−+ −
1
11
1
2 211
1ddx x a
xm
xa m
x
x
m m
∫ −
+−
+
>
++
+
cosh ( )
cosh
1
11
1
2
0
11
1
/
−−<
⎧
⎨⎪⎪
⎩⎪⎪ ∫ −
adx x a
2
1 0cosh ( )/
17.34.17. xxa
dxxm
xa
am
xa x
mm m
tanh tanh−+
−+
= + − + −∫ 11
11
2 21 1ddx∫
17.34.18. xxa
dxxm
xa m
xa x
mm m
coth coth−+
−+
= + − + −∫ 11
11
2 211
1ddx∫
17.34.19. xxa
dx
xm
xa
am
x dx
a xm
m m
sech
sech
−
+−
∫ =+
++ −
1
11
2 21 1 ∫∫ −
+−
>
+−
+ −
sech ( )
sech
1
11
2
0
11
1
x a
xm
xa m
x dx
a x
m m
/
22
1 0∫ − <
⎧
⎨⎪⎪
⎩⎪⎪ sech ( )x a/
17.34.20. xxa
dxxm
xa
am
x dx
x am
m m
csch csch−+
−= + ± + +∫ 11
1
2 21 1 ∫∫ + > − <( , )if ifx x0 0
TABLES OF SPECIAL INDEFINITE INTEGRALS
18 DEFINITE INTEGRALS
Definition of a Definite Integral
Let f(x) be defined in an interval a � x � b. Divide the interval into n equal parts of length �x = (b − a)/n. Then the definite integral of f(x) between x = a and x = b is defined as
18.1. f x dx f a x f a x x f a xna
b( ) lim{ ( ) ( ) ( )= + + + +
→∞∫ Δ Δ Δ Δ Δ2 xx f a n x x+ + + −� ( ( ) ) }1 Δ Δ
The limit will certainly exist if f(x) is piecewise continuous.
If f xddx
g x( ) ( ),= then by the fundamental theorem of the integral calculus the above definite integral can
be evaluated by using the result
18.2. f x dxddx
g x dx g x g b g aa
b
a
b
a
b( ) ( ) ( ) ( ) ( )= = = −∫ ∫
If the interval is infinite or if f(x) has a singularity at some point in the interval, the definite integral is called an improper integral and can be defined by using appropriate limiting procedures. For example,
18.3. f x dx f x dxba a
b( ) lim ( )=
→∞
∞
∫ ∫
18.4. f x dx f x dxa a
b
b
( ) lim ( )=→−∞−∞
∞
→∞∫ ∫
18.5. f x dx f x dx b( ) lim ( )=→∈ 0
if is a singular point..α
∈b
a
b −
∫∫
18.6. f x dx dx aa
b
a
b( ) lim=
→ +∫∫ ∈ ∈0f x( ) if is a singulaar point.
General Formulas Involving Definite Integrals
18.7. { ( ) ( ) ( ) } ( ) ( )f x g x h x dx f x dx g x dxa
b
a
b
a
b± ± ± = ± ±∫ ∫� ∫∫ ∫ ±h x dx
a
b( ) �
18.8. cf x dx c f x dx ca
b
a( ) ( )=∫ where is any constant.
bb
∫
18.9. f x dxa
a( ) =∫ 0
18.10. f x dx f x dxa
b
b
a( ) ( )= −∫ ∫
18.11. f x dx f x dx f x dxa
b
a
c
c
b( ) ( ) ( )= +∫ ∫ ∫
108
109
18.12. f x dx b a f c c a ba
b( ) ( ) ( ) .= −∫ where is between and
This is called the mean value theorem for definite integrals and is valid if f(x) is continuous in a � x � b.
18.13. f x g x dx f c g x dx c aa
b( ) ( ) ( ) ( )=∫ where is between and bb
a
b
∫This is a generalization of 18.12 and is valid if f(x) and g(x) are continuous in a � x � b and g(x) � 0.
Leibnitz’s Rules for Differentiation of Integrals
18.14.d
dF x dx
Fd
dx Fddα α α φ α φ
αφ α
φ α
φ( , ) ( , )
( )
( )= ∂ +∫
1
2
22
11
2
11
( )
( )( , )
α
φ αφ α φ
α∫ − Fdd
Approximate Formulas for Definite Integrals
In the following the interval from x = a to x = b is subdivided into n equal parts by the pointsa = x0, x1, x2, …, xn–1, xn = b and we let y0 = f(x0), y1 = f(x1), y2 = f(x2), …, yn = f(xn), h = (b – a)/n.
Rectangular formula:
18.15. f x dx h y y y yna
b( ) ( )≈ + + + + −∫ 0 1 2 1�
Trapezoidal formula:
18.16. f x dxh
y y y y yn na
b( ) ( )≈ + + + + +−∫ 2
2 2 20 1 2 1�
Simpson’s formula (or parabolic formula) for n even:
18.17. f x dxh
y y y y y y yn n na( ) ( )≈ + + + + + + +− −3
4 2 4 2 40 1 2 3 2 1�bb
∫
Definite Integrals Involving Rational or Irrational Expressions
18.18.dx
x a a2 20 2+ =∞
∫π
18.19.x dx
x pp
p−∞
+ = < <∫1
0 10 1
ππsin
,
18.20. x dxx a
an m n
m nm
n n
m n
+ = + < + <∞ + −
∫0
1
10 1
ππsin[( ) / ]
,
18.21.x dx
x x mmm
1 2 20 + + =∞
∫ cos sinsinsinβ
ππ
ββ
18.22.dx
a x
a
2 20 2−=∫
π
18.23. a x dxaa
2 2
0
2
4− =∫
π
DEFINITE INTEGRALS
110
18.24. x a x dxa m n p
n mm n n p
m np
( )[( )/ ] ( )
[(− = + +
++ +1 1 1
1Γ Γ
Γ ))/ ]n pa
+ +∫ 10
18.25.x dx
x aa m n
n
m
n n r
r m nr
( )( ) [( )/ ]
sin[+ = − +− + −1 11 1π Γ(( ) / ]( )! [( ) / ]
,m n r m n r
m nr+ − + − + < + <∞
∫ 1 1 1 10 1
0 π Γ
Definite Integrals Involving Trigonometric Functions
All letters are considered positive unless otherwise indicated.
18.26. sin sin,
/ ,mx nx dx
m n m n
m n=
≠0
2
integers and
integeπ rrs and m n=
⎧⎨⎪
⎩⎪∫0
π
18.27. cos cos,
/ ,mx nx dx
m n m n
m n0
0
2
π
π∫ =
≠integers and
inntegers and m n=
⎧⎨⎪
⎩⎪
18.28. sin cos,
/ (mx nx dx
m n m n
m m=
+0
2 2
integers and even
−− +
⎧⎨⎪
⎩⎪∫
n m n m n20 ) , integers and odd
π
18.29. sin cos//
2 2
0
2
0
2
4x dx x dx= =∫∫
πππ
18.30. sin cos/
2 2
0
2 1 3 5 2 12 4 6 2 2
m mx dx x dxm
m= = −∫
π πi i �i i � ,, , ,
/m =∫ 1 2
0
2…
π
18.31. sin cos/
2 1
0
22 1 2 4 6 2
1 3 5 2m mx dx x dx
mm
+ +∫ = =π i i �
i i � ++ =∫ 11 2
0
2π /, , ,m …
18.32. sin cos( ) ( )( )
/2 1 2 1
0
2
2p qx x dx
p qp q
− − = +∫Γ ΓΓ
π
18.33. sin/
/
pxx
dx
p
p
p
=
>
=
− <
⎧
⎨⎪⎪
⎩⎪⎪
∞
∫π
π
2 0
0 0
2 0
0
18.34.sin cos
/
/
px qxx
dx
p q
p q
p q
0
0 0
2 0
4 0
∞
∫ =
> >
< <
= >
⎧
⎨⎪⎪π
π⎩⎩⎪⎪
18.35.sin sin /
/
px qxx
dxp p q
q p q20
2 0
2 0=
<
>
⎧⎨⎪
⎩⎪
∞
∫π
π
�
�
18.36.sin2
20 2px
xdx
p∞
∫ = π
18.37.1
220
− =∞
∫cos pxx
dxpπ
DEFINITE INTEGRALS
111
18.38.cos cos
lnpx qx
xdx
qp
− =∞
∫0
18.39. cos cos ( )px qxx
dxq p− = −∞
∫ 20 2π
18.40.cos mxx a
dxa
e ma2 20 2+ = −
∞
∫π
18.41.x mxx a
dx e masin2 20 2+ = −
∞
∫π
18.42.sin( )
( )mx
x x adx
ae ma
2 2 20 21+ = − −
∞
∫π
18.43.dx
a b x a b+ =−∫ sin
22 20
2 ππ
18.44.dx
a b x a b+ =−∫ cos
22 20
2 ππ
18.45.dx
a b xb a
a b+ =−∫
−
coscos ( / )/
0
2 1
2 2
π
18.46.dx
a b xdx
a b xa
a b( sin ) ( cos ) ( ) /+ = + = −∫ 20
2
2 2 2 3 2
2π π00
2π
∫
18.47.dx
a x a aa
1 22
10 12 20
2
− + = − < <∫ cos,
ππ
18.48.x x dxa x a
a a asincos
( / ) ln ( ),
ln (1 2
1 1
120 − + =
+ <∫
π π
π ++ >
⎧⎨⎪
⎩⎪ 1 1/ ),a a
18.49.cos
cos, , , , ,
mx dxa x a
aa
a mm
1 2 11 0 1 22 2
2
0 − + = − < =ππ…∫∫
18.50. sin cosax dx ax dxa
2 2
00
12 2
= =∞∞
∫∫π
18.51. sin ( / ) sin ,/ax dxna
nn
nnn= >
∞
∫1
12
110Γ π
18.52. cos ( / ) cos ,/ax dxna
nn
nnn= >
∞
∫1
12
110Γ π
18.53.sin cosx
xdx
x
xdx= =
∞∞
∫∫ 00 2π
18.54.sin
( )sin ( / ),
xx
dxp p
pp = < <∞
∫π
π2 20 1
0 Γ
18.55.cos
( ) cos ( / ),
xx
dxp p
pp = < <∞
∫π
π2 20 1
0 Γ
18.56. sin cos cos sinax bx dxa
ba
ba
2
0
2 2
212 2
∞
∫ = −⎛⎝⎜
⎞⎠⎟
π
DEFINITE INTEGRALS
112
18.57. cos cos cos sinax bx dxa
ba
ba
22 2
02
12 2
= +⎛⎝⎜
⎞⎠⎟
∞
∫π
18.58.sin3
30
38
xx
dx =∞
∫π
18.59.sin4
40 3x
xdx =
∞
∫π
18.60.tan x
xdx =
∞
∫π20
18.61.dx
xm1 40
2
+ =∫ tan/ ππ
18.62. xx
dxsin
/= − + − +{ }∫ 2
11
13
15
172 2 2 20
2�
π
18.63. tan−
∫ = − + − +1
0
1
2 2 2 2
11
13
15
17
xx
dx �
18.64.sin
ln−
∫ =1
0
1
22
xx
dxπ
18.65.1
0
1
1
− − =∫ ∫∞cos cosx
xdx
xx
dx γ
18.66.1
1 20 +−
⎛⎝⎜
⎞⎠⎟
=∞
∫ xx
dxx
cos γ
18.67.tan tan
ln− −∞ − =∫
1 1
0 2px qx
xdx
pq
π
Definite Integrals Involving Exponential Functions
Some integrals contain Euler’s constant g = 0.5772156 . . . (see 1.3, page 3).
18.68. e bx dxa
a bax−
∞= +∫ cos 2 20
18.69. e bx dxb
a bax−
∞= +∫ sin 2 20
18.70. e bxx
dxba
ax−∞−∫ =sin
tan0
1
18.71. e ex
dxba
ax bx− −∞ − =∫0ln
18.72. e dxa
ax−∞
=∫ 2 120
π
18.73. e bx dxa
eax b a− −∞
=∫ 2 212
4
0cos /π
DEFINITE INTEGRALS
113
18.74. e dxa
eb
aax bx c b ac a− + + −
∞=∫ ( ) ( )/2 21
2 24 4
0
πerfc
where erfc (p) = −∞
∫2 2
πe dxx
p
18.75. e dxa
eax bx c b ac a− + + −−∞
∞=∫ ( ) ( )/2 2 4 4π
18.76. x e dxna
n axn
−+
∞= +∫
Γ( )110
18.77. x e dxma
m axm
−+
∞= +∫ 2 1 2
2 1 20
Γ[( )/ ]( )/
18.78. e dxa
eax b x ab− +∞
−=∫ ( / )2 2 120
2π
18.79.x dx
ex − = + + + + =∞
∫ 111
12
13
14 60 2 2 2 2
2
�π
18.80.x
edx n
n
x n n n
−∞
− = + + +⎛⎝⎜
⎞⎠⎟∫
1
0 111
12
13
Γ( ) �
For even n this can be summed in terms of Bernoulli numbers (see pages 142–143).
18.81.x dx
ex + = − + − + =∞
∫ 111
12
13
14 120 2 2 2 2
2
�π
18.82.x
edx n
n
x n n n
−∞
+ = − + −⎛⎝⎜
⎞⎠⎟∫
1
0 111
12
13
Γ( ) �
For some positive integer values of n the series can be summed (see 23.10).
18.83.sin
cothmx
edx
mmx20 1
14 2
12π − = −
∞
∫
18.84.1
10 + −⎛⎝⎜
⎞⎠⎟ =−
∞
∫ xe
dxx
x γ
18.85.e e
xdx
x x− −∞ − =∫2
0
12 γ
18.86.1
10 eex
dxx
x
− −⎛⎝⎜
⎞⎠⎟
=−∞
∫ γ
18.87.e ex px
dxb pa p
ax bx− −∞ − = ++
⎛⎝⎜
⎞⎠⎟∫ sec
ln0
2 2
2 2
12
18.88.e ex px
dxbp
ap
ax bx− −∞− −− = −∫ csc
tan tan0
1 1
18.89. e xx
dx aa
aax−∞
−− = − +∫( cos )
cot ln ( )1
2120
1 2
DEFINITE INTEGRALS
114
Definite Integrals Involving Logarithmic Functions
18.90. x x dxn
mm nm n
n
n(ln )( ) !
( ), , , ,= −
+ > − =+∫1
11 0 1 210
1…
If n ≠ 0, 1, 2, … replace n! by Γ(n + 1).
18.91.ln x
xdx
1 12
2
0
1
+ = −∫π
18.92.ln x
xdx
1 6
2
0
1
− = −∫π
18.93.ln ( )1
120
1 2+ =∫x
xdx
π
18.94.ln ( )1
60
1 2− = −∫x
xdx
π
18.95. ln ln ( ) lnx x dx1 2 2 212
2
0
1+ = − −∫
π
18.96. ln ln ( )x x dx1 26
2
0
1− = −∫
π
18.97.x x
xdx p p p
p−∞
+ = − < <∫1
0
2
10 1
lncsc cotπ π π
18.98.x x
xdx
mn
m n− = ++∫ ln
ln0
1 11
18.99. e x dxx−∞
= −∫ ln γ0
18.100. e x dxx−∞
= − +∫ 2
42 2
0ln ( ln )
π γ
18.101. lnee
dxx
x
+−
⎛⎝⎜
⎞⎠⎟
=∞
∫11 40
2π
18.102. ln sin ln cos ln//
x dx x dx= = −∫∫πππ
22
0
2
0
2
18.103. (ln sin ) (ln cos ) (ln )/
x dx x dx2 2 23
0
2
0 22
24= = +∫
π ππππ /2
∫
18.104. x x dxln sin ln= −∫ππ 2
0 22
18.105. sin ln sin ln/
x x dx = −∫ 2 10
2π
18.106. ln ( sin ) ln ( cos ) ln ( )a b x dx a b x dx a a b+ = + = + −2 2 2
0
2π
πππ
∫∫0
2
18.107. ln( cos ) lna b x dxa a b+ = + −⎛
⎝⎜⎞
⎠⎟∫ ππ 2 2
0 2
DEFINITE INTEGRALS
115
18.108. ln ( cos )ln ,
ln ,a ab x b dx
a a b
b b a2 22
2 0
2 0− + =
>
>
⎧ π
π
�
�⎨⎨⎪
⎩⎪∫0
π
18.109. ln ( tan ) ln/
18
20
4+ =∫ x dx
ππ
18.110. sec lncoscos
{(cos ) (xb xa x
dx a11
12
1 2++
⎛⎝⎜
⎞⎠⎟
= −− ccos ) }/
−∫ 1 2
0
2b
π
18.111. ln sinsin sin sin
22 1
22
332 2 2
xdx
a a a⎛⎝⎜
⎞⎠⎟ = − + + +⎛ �⎝⎝⎜
⎞⎠⎟∫0
a
See also 18.102.
Definite Integrals Involving Hyperbolic Functions
18.112.sinsinh
tanhaxbx
dxb
ab
=∞
∫π π2 20
18.113.coscosh
axbx
dxb
ab
=∞
∫π π2 20
sech
18.114.x dx
ax asinh0
2
24∞
∫ = π
18.115.x dx
ax an
n n
n n n nsinh( )
0
1
1 1 1
2 12
11
11
2∞ +
+ + +∫ = − + +Γ ++ +{ }+1
3 1n �
If n is an odd positive integer, the series can be summed.
18.116.sinh
cscax
edx
bab abx + = −
∞
∫ 1 21
20
π π
18.117.sinh
cotax
edx
a babbx − = −
∞
∫ 11
2 20
π π
Miscellaneous Definite Integrals
18.118.f ax f bx
xdx f f
ba
( ) ( ){ ( ) ( )}ln
− = − ∞∞
∫00
This is called Frullani’s integral. It holds if f ′(x) is continuous and f x f
xdx
( ) ( )− ∞∞
∫0 converges.
18.119.dxx x = + + +∫
11
12
131 20
1
3 �
18.120. ( ) ( ) ( )( ) ( )( )
a x a x dx am nm n
m n m n+ − = +− − + −
−1 1 12
Γ ΓΓaa
a
∫
DEFINITE INTEGRALS
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