Download Section 7.2 Trigonometric Integrals

7.2
Trigonometric Integrals
Trig Identities
Basic identities
sin 2 x  cos2 x  1
tan 2 x  1 sec2 x
cot 2 x  1  csc2 x
Half-angle identities
1
1
2
2
sin x  (1  cos 2 x)
cos x  (1  cos 2 x)
2
2
Product-to-Sum identities
sin A cos B  12 [sin( A  B)  sin( A  B)]
cos A cos B  12 [cos( A  B)  cos( A  B)]
sin A sin B  12 [cos( A  B)  cos( A  B)]
Examples
 cos
3
4
x sin x dx
6
4
tan
x
sec
x dx

7
4
sin
x
cos
x dx

3
5
tan
x
sec
x dx

4
sin
 x dx
 sin( 5 x) cos(3x) dx
 6 sin x cos x dx

3 sin x
1  cos x
2
dx
Strategy
1)
m
n
sin
x
cos
x dx

 If one of the power is odd, save one sine (or cosine) factor, and express the
remaining expression in terms of cosine (or sine). Then use substitution.
 If both powers are even, use half-angle identities to reduce the powers
2)
m
n
tan
x
sec
x dx

 If power of secant is even, save a factor of sec2x and express the
remaining expression in terms of tan x. Then use substitution.
 If power of tangent is odd, save a factor of (sec x tan x) and express the
remaining expression in terms of sec x. Then use substitution.
3)
 sin mx cos nx
or  sin mx sin nx or
 Use product-to-sum identities.
 cos mx cos nx