02 Trigonometric Integrals
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Transcript of 02 Trigonometric Integrals
7/27/2019 02 Trigonometric Integrals
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Trig Integrals Trig Integrals Trig Integrals Exercises
Techniques of Integration–TrigonometricIntegrals
Mathematics 54–Elementary Analysis 2
Institute of Mathematics
University of the Philippines-Diliman
1/26
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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 sin x
= (1−cos2 x ) sin x
= sin x −cos2x sin x
Thus,
sin3
x dx =
sin x − cos 2
x sin x
dx .
Let u = cos x , du =−sin x dx . Therefore,
sin3
x dx =−
1−u
2
du =−u + 1
3u
3+C =−cos x + 1
3cos3
x +C
3/26
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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 odd
split off a factor of sin x
express the rest of the factors in terms of cos x , using sin2
x = 1−cos2x
use the substitution u = cos x , du =−sin x dx
m is even
use the half-angle identity
sin2x = 1
2(1−cos2x )
4/26
m
m
m n
7/27/2019 02 Trigonometric Integrals
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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 odd
split off a factor of cos x
express the rest of the factors in terms of sin x , using cos2
x = 1−sin2x
use the substitution u = sin x , du = cos x dx
m is even
use the half-angle identity
cos2x = 1
2(1+cos2x )
5/26
T i I l T i I l T i I l E i
i m d
m d
i m n d
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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 =
cos4x cos x dx
= cos2x
2cos x dx
=1
−sin2
x 2
cos x dx
=
1−2sin 2x +sin4
x
cos x dx
Let u = sin x , du = cos x dx .
cos5 x dx =
1−2u 2+u 4
du
= u − 2
3u
3+ 1
5u
5+C
= sin x − 2
3sin3
x + 1
5sin5
x +C
6/26
T i I t l T i I t l T i I t l E i
i m d
m d
i m n d
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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 sin2x dx .
cos
3
x sin2
x dx =cos2 x sin2 x cos x dx
=
1−sin2x
sin2x cos x dx
=
sin2x cos x dx −
sin4
x cos x dx
let u = sin x du = cos xdx cos3
x sin2x dx =
u
2du −
u
4du
=1
3
u 3
−1
5
u 5
+C
=1
3
sin3x
−1
5
sin5x
+C
7/26
Trig Integrals Trig Integrals Trig Integrals Exercises
sinm x dx or
cosm x dx
sinm x cosn x dx
7/27/2019 02 Trigonometric Integrals
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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 sin x
express the rest of the factors in terms of cos x , using
sin2x = 1−cos2
x
use the substitution u = cos x , du =−sin x dx
8/26
Trig Integrals Trig Integrals Trig Integrals Exercises
sinm x dx or
cosm x dx
sinm x cosn x dx
7/27/2019 02 Trigonometric Integrals
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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 odd
split off a factor of cos x
express the rest of the factors in terms of sin x , using cos2
x = 1−sin2x
use the substitution u = sin x , du = cos x dx
both m and n are even
use the half-angle identities
cos2x = 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
7/27/2019 02 Trigonometric Integrals
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Trig Integrals Trig Integrals Trig Integrals Exercises
sin x dx or
cos x dx
sin x cos x dx
Example.
Evaluate sin2x cos4
x dx .
sin2
x cos4x dx =
sin2
x (cos2x )2
dx
=1
−cos2x
21
+cos2x
2
2
dx
=
1−cos2x
2
1+2cos2x +cos2 2x
4
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
7/27/2019 02 Trigonometric Integrals
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Trig Integrals Trig Integrals Trig Integrals Exercises
tan x dx or cot x dx
sec x dx or
csc x dx
Trigonometric IntegralsIntegrals of the form
tanm
x dx or
cotm x dx
Example.
Evaluate
tan3
x dx .
tan x tan
2
x dx = tan x
sec
2
x −1
dx
=
tan x sec2x dx −
tan x dx
let u = tan x , du = sec2x dx
tan3 x dx =
u du − ln |sec x |+C
= 1
2u
2− ln |sec x |+C
=
1
2tan2
x −ln
|sec x
|+C
11/26
Trig Integrals Trig Integrals Trig Integrals Exercises
tanm x dx or cotm x dx
secn x dx or
cscn x dx
7/27/2019 02 Trigonometric Integrals
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g g g g g g
Trigonometric IntegralsIntegrals of the form
tanm
x dx or
cotm x dx
tanm
x dx
split off a factor of tan2x and write this as tan2
x = sec2x −1
use the substitution u
=tan x , du
=sec2
x dx
cotm x dx
split off a factor of cot2x and write this as cot2
x = csc2x −1
use the substitution u = cot x , du =−csc2x dx
12/26
Trig Integrals Trig Integrals Trig Integrals Exercises
tanm x dx or cotm x dx
secn x dx or
cscn x dx
7/27/2019 02 Trigonometric Integrals
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g g g g g g
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
u
2du + 1
3cot3x +x +C
= −1
9u
3+ 1
3cot3x +x +C
= −1
9 cot
3
3x +
1
3 cot3x +
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
7/27/2019 02 Trigonometric Integrals
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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+cot2x
csc2xdx
=
(1+2cot2x +cot4
x )csc2x dx
let u = cot x ⇒ du =−csc2 x dx csc6
x dx =−
(1+2u 2+u
4) du
=−
cot x + 2cot3x
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
7/27/2019 02 Trigonometric Integrals
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Trigonometric IntegralsIntegrals of the form
secn
x dx or
cscn x dx
secn xdx
n is even
split off a factor of sec2x .
express the rest of the factors in terms of tan x , using
sec2
x = 1+ tan2
x use the substitution u = tan x , du = sec2
xdx .
cscn
xdx
n is evensplit off a factor of csc2
x .
express the rest of the factors in terms of cot x , using
csc2x = 1+cot2
x
use the substitution u
=cot x , 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
7/27/2019 02 Trigonometric Integrals
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Example.
Evaluate
sec3
x dx .
Note that sec3x = sec x sec2
x . By IBP,
u = sec x , dv = sec2x dx
du = sec x tan x dx , v = tan x dx
sec3
x dx = sec x tan x −
tan x (sec x tan x ) dx
= sec x tan x −
tan2x sec x dx
=sec x tan x
−(sec2x
−1)sec x dx
sec3x dx = sec x tan x −
sec3
x dx +
sec x dx
2
sec3
xdx = sec x tan x + ln |sec x + tan x |+C
∴
sec3
xdx = 1
2(sec x tan x + ln |sec x + tan x |)+C
16/26
Trig Integrals Trig Integrals Trig Integrals Exercises
tanm x dx or cotm x dx
secn x dx or
cscn x dx
7/27/2019 02 Trigonometric Integrals
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Trigonometric IntegralsIntegrals of the form
secn
x dx or
cscn x dx
secn
xdx
n is odd
split off a factor of sec2x
use IBP with dv = sec2 x dx and u to be the remaining factors
cscn
xdx
n is oddsplit off a factor of csc2
x
use IBP, with dv = csc2x 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
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Trigonometric IntegralsIntegrals of the form
tanm
x secn x dx or
cotm
x cscn x dx
Example.
Evaluate
tan3
x sec2x dx .
tan
3
x sec2
x dx = tan
2
x sec x sec x tan x dx
=
sec2x −1
sec x sec x tan x dx
=
sec3x −sec x sec x tan x dx
let u = sec x , du = sec x tan x dx tan3
x sec2x dx =
u
3−u
du
=
1
4
sec4x
−
1
2
sec2x
+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
7/27/2019 02 Trigonometric Integrals
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Trigonometric IntegralsIntegrals of the form
tanm
x secn x dx or
cotm
x cscn x dx
tanm x secn x dx
m is odd
split off a factor of sec x tan x
express the rest of the factors in terms of sec x using the identity
tan2
x = sec2
x −1use the substitution u = sec x , du = sec x tan x dx
cotm
x cscn x dx
m is oddsplit off a factor of csc x cot x
express the rest of the factors in terms of csc x using the identity
cot2x = csc2
x −1
use the substitution u = csc x , du =−csc x cot x 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
7/27/2019 02 Trigonometric Integrals
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Trigonometric IntegralsIntegrals of the form
tanm
x secn x dx or
cotm
x cscn x dx
tanm x secn x dx
n is even
split off a factor of sec2x
express the rest of the factors in terms of tan x using the identity
sec2
x = 1+ tan2
x use the substitution u = tan x , du = sec2
x dx
cotm
x cscn x dx
n is evensplit off a factor of csc2
x
express the rest of the factors in terms of cot x using the identity
csc2x = 1+cot2
x
use the substitution u = cot x , du =−csc2x 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
7/27/2019 02 Trigonometric Integrals
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Example.
Evaluate cot2x csc x dx .
cot2
x csc x dx =
(csc2x −1)csc x dx
=(csc3 x −csc x ) dx
=
csc3x dx − ln |csc x −cot x |
Exercise:
csc3
x dx =−1
2 csc x cot x +1
2 ln |csc x −cot x |+C
=−1
2csc x cot x − 1
2ln |csc x −cot x |+C
21/26
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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
7/27/2019 02 Trigonometric Integrals
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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 odd
express the even power of tan x in terms of sec x using the
identity tan2x = 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 cot x in terms of csc x using the
identity cot2x = 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
i i l
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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
],
sin mx sin nx = −1
2[cos(m +n )x −cos(m −n )x ],
cos mx cos nx = 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
7/27/2019 02 Trigonometric Integrals
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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
E i