02 trigonometric integrals - handout
TRANSCRIPT
Trig Integrals Trig Integrals Trig Integrals Exercises
Techniques of Integration–TrigonometricIntegrals
Mathematics 54–Elementary Analysis 2
Institute of MathematicsUniversity of the Philippines-Diliman
1 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Trigonometric IntegralsIntegrals of the form
∫sinm x dx or
∫cosm x dx
Recall∫sinx dx =−cosx+C∫sin2 x dx =
∫1
2(1−cos2x) dx = 1
2x− 1
4sin2x+C
2 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Trigonometric IntegralsIntegrals of the form
∫sinm x dx or
∫cosm x dx
Example.
Consider∫
sin3 x dx.
Note thatsin3 x = sin2 x sinx
= (1−cos2 x) sinx
= sinx−cos2 x sinx
Thus,∫
sin3 x dx =∫ (
sinx− cos 2x sinx)
dx.
Let u = cosx, du =−sinx dx. Therefore,
∫sin3 x dx =−
∫ (1−u2) du =−u+ 1
3u3 +C =−cosx+ 1
3cos3 x+C
3 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Trigonometric IntegralsIntegrals of the form
∫sinm x dx or
∫cosm x dx
∫sinm x dx, m ∈N
m is oddsplit off a factor of sinx
express the rest of the factors in terms of cosx, usingsin2 x = 1−cos2 x
use the substitution u = cosx, du =−sinx dx
m is evenuse the half-angle identity
sin2 x = 1
2(1−cos2x)
4 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Trigonometric IntegralsIntegrals of the form
∫sinm x dx or
∫cosm x dx
∫cosm x dx, m ∈N
m is oddsplit off a factor of cosx
express the rest of the factors in terms of sinx, usingcos2 x = 1− sin2 x
use the substitution u = sinx, du = cosx dx
m is evenuse the half-angle identity
cos2 x = 1
2(1+cos2x)
5 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Example.
Evaluate∫
cos5 x dx
∫cos5 x dx =
∫cos4 x cosx dx
=∫ (
cos2 x)2
cosx dx =∫ (
1− sin2 x)2
cosx dx
=∫ (
1−2sin 2x+ sin4 x)
cosx dx
Let u = sinx, du = cosx dx.∫cos5 x dx =
∫ (1−2u2 +u4) du
= u− 2
3u3 + 1
5u5 +C
= sinx− 2
3sin3 x+ 1
5sin5 x+C
6 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Trigonometric IntegralsIntegrals of the form
∫sinm x cosn x dx
Example.
Evaluate∫
cos3 x sin2 x dx.
∫cos3 x sin2 x dx = ∫
cos2 x sin2 x cosx dx
=∫ (
1− sin2 x)
sin2 x cosx dx
=∫
sin2 x cosx dx−∫
sin4 x cosx dx
let u = sinx du = cosxdx∫cos3 x sin2 x dx =
∫u2 du−
∫u4 du
= 1
3u3 − 1
5u5 +C = 1
3sin3 x− 1
5sin5 x+C
7 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Trigonometric IntegralsIntegrals of the form
∫sinm x cosn x dx
∫sinm x cosn x dx
m is oddsplit off a factor of sinx
express the rest of the factors in terms of cosx, usingsin2 x = 1−cos2 x
use the substitution u = cosx, du =−sinx dx
8 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Trigonometric IntegralsIntegrals of the form
∫sinm x cosn x dx
∫sinm x cosn x dx
n is oddsplit off a factor of cosx
express the rest of the factors in terms of sinx, usingcos2 x = 1− sin2 x
use the substitution u = sinx, du = cosx dx
both m and n are evenuse the half-angle identities
cos2 x = 1
2(1+cos2x) and sin2 x = 1
2(1−cos2x)
use the rule for∫
cosm x dx
9 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
sinm x dx or∫
cosm x dx∫
sinm x cosn x dx
Example.
Evaluate∫
sin2 x cos4 x dx.
∫sin2 x cos4 x dx =
∫sin2 x (cos2 x)2dx
=∫ (
1−cos2x
2
)(1+cos2x
2
)2
dx
=∫ (
1−cos2x
2
)(1+cos2x
2
)2
dx
= 1
8
∫ (1+cos2x−cos2 2x−cos3 2x
)dx
= 1
8
∫ [1+cos2x−
(1+cos4x
2
)− (1− sin2 2x)cos2x
]dx
= 1
8
[x+ sin2x
2− 1
2
(x+ sin4x
4
)− 1
2
(sin2x− sin3 2x
3
)]+C
10 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x dx or∫
cotm x dx∫
secn x dx or∫
cscn x dx
Trigonometric IntegralsIntegrals of the form
∫tanm x dx or
∫cotm x dx
Example.
Evaluate∫
tan3 x dx.
∫tanx tan2 x dx =
∫tanx
(sec2 x−1
)dx
=∫
tanx sec2 x dx−∫
tanx dx
let u = tanx, du = sec2 x dx∫tan3 x dx =
∫udu− ln |secx|+C
= 1
2u2 − ln |secx|+C
= 1
2
(tan2 x
)− ln |secx|+C
11 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x dx or∫
cotm x dx∫
secn x dx or∫
cscn x dx
Trigonometric IntegralsIntegrals of the form
∫tanm x dx or
∫cotm x dx
∫tanm x dx
split off a factor of tan2 x and write this as tan2 x = sec2 x−1
use the substitution u = tanx, du = sec2 x dx
∫cotm x dx
split off a factor of cot2 x and write this as cot2 x = csc2 x−1
use the substitution u = cotx, du =−csc2 x dx
12 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x dx or∫
cotm x dx∫
secn x dx or∫
cscn x dx
Example.
Evaluate∫
cot4 3x dx.
∫cot2 3x cot2 3x dx =
∫cot2 3x
(csc2 3x−1
)dx
=∫ (
cot2 3x csc2 3x−cot2 3x)
dx
=∫ (
cot2 3x csc2 3x−csc2 3x+1)
dx
=∫ (
cot2 3x csc2 3x)
dx+ 1
3cot3x+x+C
let u = cot3x, du =−3csc2 3x dx∫cot4 3x dx = −1
3
∫u2 du+ 1
3cot3x+x+C
= −1
9u3 + 1
3cot3x+x+C
= −1
9cot3 3x+ 1
3cot3x+x+C
13 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x dx or∫
cotm x dx∫
secn x dx or∫
cscn x dx
Trigonometric IntegralsIntegrals of the form
∫secn x dx or
∫cscn x dx
Example.
Evaluate∫
csc6 x dx.
∫csc6 x dx =
∫(csc2 x)2 csc2 x dxdx
=∫ (
1+cot2 x)
csc2 xdx
=∫
(1+2cot2 x+cot4 x)csc2 x dx
let u = cotx ⇒ du =−csc2 x dx∫csc6 x dx =−∫
(1+2u2 +u4) du
=−(cotx+ 2cot3 x
3+ cot5 x
5
)+C
14 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x dx or∫
cotm x dx∫
secn x dx or∫
cscn x dx
Trigonometric IntegralsIntegrals of the form
∫secn x dx or
∫cscn x dx
∫secn xdx
n is evensplit off a factor of sec2 x.express the rest of the factors in terms of tanx, usingsec2 x = 1+ tan2 xuse the substitution u = tanx, du = sec2 xdx.
∫cscn xdx
n is evensplit off a factor of csc2 x.express the rest of the factors in terms of cotx, usingcsc2 x = 1+cot2 xuse the substitution u = cotx, du =−csc2 xdx
15 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x dx or∫
cotm x dx∫
secn x dx or∫
cscn x dx
Example.
Evaluate∫
sec3 x dx.
Note that sec3 x = secx sec2 x. By IBP,
u = secx , dv = sec2 x dxdu = secx tanx dx , v = tanx dx
∫sec3 x dx = secx tanx−
∫tanx(secx tanx) dx
= secx tanx−∫
tan2 x secx dx
= secx tanx−∫
(sec2 x−1)secx dx∫sec3 x dx = secx tanx−
∫sec3 x dx+
∫secx dx
2∫
sec3 xdx = secx tanx+ ln |secx+ tanx|+C
∴∫
sec3 xdx = 1
2(secx tanx+ ln |secx+ tanx|)+C
16 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x dx or∫
cotm x dx∫
secn x dx or∫
cscn x dx
Trigonometric IntegralsIntegrals of the form
∫secn x dx or
∫cscn x dx
∫secn xdx
n is oddsplit off a factor of sec2 xuse IBP with dv = sec2 x dx and u to be the remaining factors
∫cscn xdx
n is oddsplit off a factor of csc2 xuse IBP, with dv = csc2 x dx and u to be the remaining factors
17 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Trigonometric IntegralsIntegrals of the form
∫tanm x secn x dx or
∫cotm x cscn x dx
Example.
Evaluate∫
tan3 x sec2 x dx.
∫tan3 x sec2 x dx =
∫tan2 x secx secx tanx dx
=∫ (
sec2 x −1)
secx secx tanx dx
=∫ (
sec3 x− secx)
secx tanx dx
let u = secx, du = secx tanx dx∫tan3 x sec2 x dx =
∫ (u3 −u
)du
= 1
4sec4 x− 1
2sec2 x+C
18 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Trigonometric IntegralsIntegrals of the form
∫tanm x secn x dx or
∫cotm x cscn x dx
∫tanm x secn x dx
m is oddsplit off a factor of secx tanxexpress the rest of the factors in terms of secx using the identitytan2 x = sec2 x−1use the substitution u = secx, du = secx tanx dx
∫cotm x cscn x dx
m is oddsplit off a factor of cscx cotxexpress the rest of the factors in terms of cscx using the identitycot2 x = csc2 x−1use the substitution u = cscx, du =−cscx cotx dx
19 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Trigonometric IntegralsIntegrals of the form
∫tanm x secn x dx or
∫cotm x cscn x dx
∫tanm x secn x dx
n is evensplit off a factor of sec2 xexpress the rest of the factors in terms of tanx using the identitysec2 x = 1+ tan2 xuse the substitution u = tanx, du = sec2 x dx
∫cotm x cscn x dx
n is evensplit off a factor of csc2 xexpress the rest of the factors in terms of cotx using the identitycsc2 x = 1+cot2 xuse the substitution u = cotx, du =−csc2 x dx
20 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Example.
Evaluate∫
cot2 x cscx dx.
∫cot2 x cscx dx =
∫(csc2 x−1)cscx dx
=∫
(csc3 x−cscx) dx
=∫
csc3 x dx− ln |cscx−cotx|
Exercise:∫
csc3 x dx =−1
2cscx cotx+ 1
2ln |cscx−cotx|+C
=−1
2cscx cotx− 1
2ln |cscx−cotx|+C
21 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Example.
Evaluate∫ p
tanx sec4 x dx.
∫ ptanx sec4 x dx =
∫ ptanx sec2 x sec2 x dx
=∫ p
tanx(1+ tan2 x)sec2 x dx
=∫ (p
tanx+√
tan5 x)
sec2 x dx
= 2
3
√tan3 x+ 2
7
√tan7 x+C
22 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Trigonometric IntegralsIntegrals of the form
∫tanm x secn x dx or
∫cotm x cscn x dx
∫tanm x secn x dx
m is even and n is oddexpress the even power of tanx in terms of secx using theidentity tan2 x = sec2 x−1
use the rule for∫
secm x dx
∫cotm x cscn x dx
m is even and n is oddexpress the even power of cotx in terms of cscx using theidentity cot2 x = csc2 x−1
use the rule for∫
cscm x dx
23 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Trigonometric IntegralsF. Integrals of the form
∫sin mx cos nxdx,
∫sin mx sin nxdx or
∫cos mx cos nxdx
Recall. Product to Sum Formula
sinmx cosnx = 1
2[sin(m+n)x+ sin(m−n)x],
sinmx sinnx = −1
2[cos(m+n)x−cos(m−n)x],
cosmx cosnx = 1
2[cos(m+n)x+cos(m−n)x].
24 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises∫
tanm x secn x dx or∫
cotm x cscn x dx∫
sin mx cos nx dx,∫
sin mx sin nx dx or∫
cos mx cos nx dx
Example.
Evaluate∫
cos3x cos5x dx.
∫cos3x cos5x dx = 1
2
∫(cos(3x+5x)+cos(3x−5x)) dx
= 1
2
∫(cos8x+cos2x) dx
= 1
2
(1
8sin8x+ 1
2sin2x
)+C
= 1
16sin8x+ 1
4sin2x+C
25 / 26
Trig Integrals Trig Integrals Trig Integrals Exercises
Exercises
Evaluate the following integrals.
1
∫ 1
0sin2πx cos2πx dx
2
∫cos3 xp
sinxdx
3
∫csc4 x
cot2 xdx
4
∫cos 4x cos 3x dx
5
∫tan3(lnx)sec8(lnx)
xdx
26 / 26