Download Sec 7.2

Sec 7.2: TRIGONOMETRIC INTEGRALS
Example
Find  cos x dx
3
3
2
cos
x
dx

cos
x cos xdx


  (1  sin 2 x) cos xdx
Example
Find
5
2
sin
x
cos
x dx

5
2
4
2
sin
x
cos
x
dx

sin
x
cos
x sin x dx


  (1  cos 2 x) 2 cos 2 x sin xdx
Sec 7.2: TRIGONOMETRIC INTEGRALS
 sin
m
n
x cos x dx
sin  odd
m is odd
cos  odd
n is odd
save one sin
1 save one cos with dx
1
2 use cos x  1- sin x
2 use sin x  1- cos x
2
2
to express the remaining
factors in terms of sin
3
2
cos
x
dx

cos
x cos xdx


2
2
to express the remaining
factors in terms of cos
5
2
sin
x
cos
x dx

  sin 4 x cos 2 x sin x dx
Sec 7.2: TRIGONOMETRIC INTEGRALS
sin  even
cos  even
4
sin
 dx

1
4
2


1

cos
2
x
dx

1
sin
cos
odd
even
even
odd
odd
odd
even
even
use half angle
sin 2 x  12 (1- cos 2 x)
2
cos2 x  12 (1  cos 2 x)
sometimes helpful to use
sin x cos x  12 sin 2 x
Sec 7.2: TRIGONOMETRIC INTEGRALS
We can use a similar strategy to
evaluate integrals of the form
 tan
m
n
x sec x dx
Example
Find
 tan
6
4
x sec x dx
u  tan x  du  sec 2 xdx
sec 2 x  1  tan 2 x
Example
Find
5
4
tan
x
sec
x dx

u  sec x  du  sec x tan xdx
tan 2 x  1  sec 2 x
Sec 7.2: TRIGONOMETRIC INTEGRALS
 tan
m
n
x sec x dx
tan  odd
m is odd
sec  even
n is even
1 save one sec 2
2 use sec x  1  tan x
2
to express the remaining
factors in terms of tan
2
1
save one sec x tan x
2
use tan 2 x  sec 2 x  1
to express the remaining
factors in terms of sec
Sec 7.2: TRIGONOMETRIC INTEGRALS
tan
even
tan
sec
odd
even
even
odd
odd
odd
even
even
sec
odd
the guidelines are not as clear-cut. We may need to use identities,
integration by parts, and occasionally a little ingenuity.
Sec 7.2: TRIGONOMETRIC INTEGRALS
tan
even
sec
odd
the guidelines are not as clear-cut. We may
need to use identities, integration by parts, and
occasionally a little ingenuity.
Example
Find
3
sec
 xdx
Powers of sec x may require integration by
parts, as shown in the following example.
Example
Find
If an even power of tangent appears with an
odd power of secant, it is helpful to express
the integrand completely in terms of sec x
 tan
3
xdx
Sec 7.2: TRIGONOMETRIC INTEGRALS
REMARK
Integrals of the form
m
n
cot
x
csc
x dx

can be found by similar methods because
of the identity
1  cot 2 x  csc 2 x
cot
x
csc
x
dx

m
n
cot  odd
m is odd
csc  even
n is even
1 save one sec 2
2
csc x  1  cot x
2
2
to express the remaining
factors in terms of cot
1
save one csc x cot x
2
use cot 2 x  csc 2 x  1
to express the remaining
factors in terms of csc
Sec 7.2: TRIGONOMETRIC INTEGRALS
 cos mx cos nx
 sin mx sin nx
dx
 sin mx cos nx
Example
Find
 sin 4 x cos 5 x
dx
dx
dx
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