Download Math 152 Class Notes September 24, 2015 8.2 Trigonometric Integrals

Math 152 Class Notes
September 24, 2015
8.2 Trigonometric Integrals
In this section we use trigonometric identities to integrate certain combinations of
trigonometric functions.
First, we recall the trig identities and integral formulas needed in this section:
ˆ
2
sin x + cos2 x = 1
sin xdx = − cos x + C
ˆ
tan2 x + 1 = sec2 x
cos xdx = sin x + C
ˆ
cot2 x + 1 = csc2 x
sec2 xdx = tan x + C
ˆ
1
sin x = (1 − cos 2x)
2
1
cos2 x = (1 + cos 2x)
2
2
sec x tan xdx = sec x + C
ˆ
csc2 xdx = − cot x + C
ˆ
csc x cot xdx = − csc x + C
ˆ
1.
Integrals of the form
a) If the power of
cos x
sinm x cosn xdx
is odd, save a factor of
express the remaining factors in terms of
ˆ
Example 1.
sin4 x cos3 xdx
sin x.
cos x
Then
cos2 x = 1 − sin2 x
substitute u = sin x.
and use
to
b) If the power of
sin x
is odd, save a factor of
express the remaining factors in terms of
sin x.
sin x
Then
sin2 x = 1 − cos2 x
substitute u = cos x.
and use
to
ˆ
sin3 xdx
Example 2.
2
c) If the powers of both sin x and cos x are even, use the identities sin
1
cos2 x = (1 + cos 2x).
2
ˆ π
Example 3.
sin2 xdx
and
0
1
x = (1−cos 2x)
2
ˆ
π/2
sin2 x cos2 xdx
Example 4.
0
ˆ
2.
Integrals of the form
a) If the power of
sec x
tanm x secn xdx
is even, save a factor of
express the remaining factors in terms of
ˆ
Example 5.
tan6 x sec4 xdx
tan x.
sec2 x
Then
sec2 x = tan2 x + 1
substitute u = tan x.
and use
to
b) If the power of
tan x
sec x tan x and use tan2 x = sec2 x − 1
sec x. Then substitute u = sec x.
is odd, save a factor of
to express the remaining factors in terms of
ˆ
tan5 x sec7 xdx
Example 6.
c) For other cases, we may need to use identities, substitution and integration by parts.
ˆ
Example 7.
tan xdx
ˆ
Example 8.
sec xdx