integration formulas
TRANSCRIPT
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Integration Formulas
1. Common Integrals
Indefinite Integral
Method of substitution
( ( )) ( ) ( )f g x g x dx f u du′ =∫ ∫
Integration by parts
( ) ( ) ( ) ( ) ( ) ( )f x g x dx f x g x g x f x dx′ ′= −∫ ∫
Integrals of Rational and Irrational Functions
1
1
nn x
x dx Cn
+
= ++∫
1lndx x C
x= +∫
c dx cx C= +∫
2
2
xxdx C= +∫
32
3
xx dx C= +∫
2
1 1dx C
x x= − +∫
2
3
x xxdx C= +∫
2
1arctan
1dx x C
x= +
+∫
2
1arcsin
1dx x C
x= +
−∫
Integrals of Trigonometric Functions
sin cosx dx x C= − +∫
cos sinx dx x C= +∫
tan ln secx dx x C= +∫
sec ln tan secx dx x x C= + +∫
( )2 1sin sin cos
2x dx x x x C= − +∫
( )2 1cos sin cos
2x dx x x x C= + +∫
2tan tanx dx x x C= − +∫
2sec tanx dx x C= +∫
Integrals of Exponential and Logarithmic Functions
ln lnx dx x x x C= − +∫
( )
1 1
2ln ln
1 1
n nn x x
x x dx x Cn n
+ +
= − ++ +
∫
x xe dx e C= +∫
ln
xx b
b dx Cb
= +∫
sinh coshx dx x C= +∫
cosh sinhx dx x C= +∫
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2. Integrals of Rational Functions
Integrals involving ax + b
( )( )
( )( )
1
11
nn ax b
ax b dxa
fo nn
r
++
+ =+
≠ −∫
1 1lndx ax b
ax b a= +
+∫
( )( )
( )( )( ) ( )
1
2
11
2,
12
n na n x bx ax b dx ax b
a n nfor n n
+≠ −
+ −+ = +
+ +≠ −∫
2ln
x x bdx ax b
ax b a a= − +
+∫
( ) ( )2 2 2
1ln
x bdx ax b
a ax b aax b= + +
++∫
( )
( )
( )( )( )( )
12
12
1,
21
n n
a n x bxdx
ax b a n nfor n
ax bn
−≠
− −=
+−
+ − −≠ −∫
( )( )
222
3
12 ln
2
ax bxdx b ax b b ax b
ax b a
+ = − + + + +
∫
( )
2 2
2 3
12 ln
x bdx ax b b ax b
ax baax b
= + − + − ++
∫
( ) ( )
2 2
3 3 2
1 2ln
2
x b bdx ax b
ax baax b ax b
= + + − ++ +
∫
( )
( ) ( ) ( )( )
3 2 122
3
21
3 2 11, 2,3
n n n
n
ax b b a b b ax bxdx
n nfo
nar n
ax b
− − − + + + = − + − − − −+
≠∫
( )1 1
lnax b
dxx ax b b x
+= −
+∫
( )2 2
1 1ln
a ax bdx
bx xx ax b b
+= − +
+∫
( ) ( )2 2 2 32
1 1 1 2ln
ax bdx a
xb a xb ab x bx ax b
+= − + −
++ ∫
Integrals involving ax2 + bx + c
2 2
1 1 xdx arctg
a ax a=
+∫
2 2
1ln
1 2
1ln
2
a xfor x a
a a xdx
x ax afor x a
a x a
−< +
= −− >
+
∫
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2
2 2
22
2 2 2
2
2 2arctan 4 0
4 4
1 2 2 4ln 4 0
4 2 4
24 0
2
ax bfor ac b
ac b ac b
ax b b acdx for ac b
ax bx c b ac ax b b ac
for ac bax b
+− >
− −
+ − −= − <
+ + − + + −− − = +
∫
2
2 2
1ln
2 2
x b dxdx ax bx c
a aax bx c ax bx c= + + −
+ + + +∫ ∫
( )
2 2
2 2
2 2
2 2 2
2 2
2 2ln arctan 4 0
2 4 4
2 2ln arctanh 4 0
2 4 4
2ln 4 0
2 2
m an bm ax bax bx c for ac b
a a ac b ac b
mx n m an bm ax bdx ax bx c for ac b
aax bx c a b ac b ac
m an bmax bx c for ac b
a a ax b
− ++ + + − >
− −+ − +
= + + + − <+ + − −
− + + − − =
+
∫
( ) ( )( )( )
( )
( )( ) ( )1 122 2 2 2
2 3 21 2 1
1 41 4n n n
n aax bdx dx
n ac bax bx c n ac b ax bx c ax bx c− −
−+= +
− −+ + − − + + + +∫ ∫
( )
2
2 22
1 1 1ln
2 2
x bdx dx
c cax bx c ax bx cx ax bx c= −
+ + + ++ +∫ ∫
3. Integrals of Exponential Functions
( )2
1cx
cx exe dx cx
c= −∫
22
2 3
2 2cx cx x xx e dx e
c c c
= − +
∫
11n cx n cx n cxnx e dx x e x e dx
c c
−= −∫ ∫
( )
1
ln!
icx
i
cxedx x
x i i
∞
=
= +⋅
∑∫
( )1
ln lncx cx
ie xdx e x E cxc
= +∫
( )2 2
sin sin coscx
cx ee bxdx c bx b bx
c b= −
+∫
( )2 2
cos cos sincx
cx ee bxdx c bx b bx
c b= +
+∫
( )( )1
2
2 2 2 2
1sinsin sin cos sin
cx ncx n cx n
n ne xe xdx c x n bx e dx
c n c n
−−
−= − +
+ +∫ ∫
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4. Integrals of Logarithmic Functions
ln lncxdx x cx x= −∫
ln( ) ln( ) ln( )b
ax b dx x ax b x ax ba
+ = + − + +∫
( ) ( )2 2
ln ln 2 ln 2x dx x x x x x= − +∫
( ) ( ) ( )1
ln ln lnn n n
cx dx x cx n cx dx−
= −∫ ∫
( )
2
lnln ln ln
ln !
i
n
xdxx x
x i i
∞
=
= + +⋅
∑∫
( ) ( )( ) ( )( )
1 11
1
1ln 1 ln lnn n n
for ndx x dx
nx n x x− −
= − +−−
≠∫ ∫
( )( )1
2
ln 1n
11l
1
m m xx xdx x
m mfor m
+ = − + +
≠∫
( )( )
( ) ( )1
1lnln
1 11ln
nmn nm m
x x nx x dx x x dx
mr
mfo m
+−
= − ≠+ +∫ ∫
( ) ( )( )
1ln ln
11
n nx x
dx for nx n
+
= ≠+∫
( )( )
2
lnln0
2
nn xx
dx for nx n
= ≠∫
( ) ( )( )
1 2 1
ln ln 1
1 11
m m m
x xdx
x m x mfor
xm
− −= − −
− −≠∫
( ) ( )
( )
( )( )
1
1
ln ln n1
l
11
n n n
m m m
x x xndx dx
mx m x xfor m
−
−= − +
−−≠∫ ∫
ln lnln
dxx
x x=∫
( )( ) ( )
1
1 lnln ln 1
!ln
i ii
ni
n xdxx
i ix x
∞
=
−= + −
⋅∑∫
( ) ( )( )( )
1
1
ln 1 ln1
n n
dx
x x nf
xor n
−= −
−≠∫
( ) ( )2 2 2 2 1ln ln 2 2 tan
xx a dx x x a x a
a
−+ = + − +∫
( ) ( ) ( )( )sin ln sin ln cos ln2
xx dx x x= −∫
( ) ( ) ( )( )cos ln sin ln cos ln2
xx dx x x= +∫
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5. Integrals of Trig. Functions
sin cosxdx x= −∫
cos sinxdx x= −∫
2 1sin sin 2
2 4
xxdx x= −∫
2 1cos sin 2
2 4
xxdx x= +∫
3 31sin cos cos
3xdx x x= −∫
3 31cos sin sin
3xdx x x= −∫
ln tansin 2
dx xxdx
x=∫
ln tancos 2 4
dx xxdx
x
π = +
∫
2cot
sin
dxxdx x
x= −∫
2tan
cos
dxxdx x
x=∫
3 2
cos 1ln tan
sin 2sin 2 2
dx x x
x x= − +∫
3 2
sin 1ln tan
2 2 4cos 2cos
dx x x
x x
π = + +
∫
1sin cos cos 2
4x xdx x= −∫
2 31sin cos sin
3x xdx x=∫
2 31sin cos cos
3x xdx x= −∫
2 2 1sin cos sin 4
8 32
xx xdx x= −∫
tan ln cosxdx x= −∫
2
sin 1
coscos
xdx
xx=∫
2sin
ln tan sincos 2 4
x xdx x
x
π = + −
∫
2tan tanxdx x x= −∫
cot ln sinxdx x=∫
2
cos 1
sinsin
xdx
xx= −∫
2cos
ln tan cossin 2
x xdx x
x= +∫
2cot cotxdx x x= − −∫
ln tansin cos
dxx
x x=∫
2
1ln tan
sin 2 4sin cos
dx x
xx x
π = − + +
∫
2
1ln tan
cos 2sin cos
dx x
xx x= +∫
2 2tan cot
sin cos
dxx x
x x= −∫
( )
( )
( )
( )2 2
sin sinsin sin
2 2
m n x m n xmx nxdx
n m nm n
m
+ −− +
+ −≠=∫
( )
( )
( )
( )2 2
cos cossin cos
2 2
m n x m n xmx nxdx
n m nm n
m
+ −− −
+ −≠=∫
( )
( )
( )
( )2 2
sin sincos cos
2 2
m n x m n xmx nxdx
m n m nm n
+ −= +
+ −≠∫
1cos
sin cos1
nn x
x xdxn
+
= −+∫
1sin
sin cos1
nn x
x xdxn
+
=+∫
2arcsin arcsin 1xdx x x x= + −∫
2arccos arccos 1xdx x x x= − −∫
( )21arctan arctan ln 1
2xdx x x x= − +∫
( )21arccot arc cot ln 1
2xdx x x x= + +∫