8.3 Trigonometric Integrals
In this section we will be solving integrals of the form:
! sin
m x cos n x dx
The following trig identities will prove quite useful in this section.
Pythagorean Identity
sin2 x + cos2 x = 1
1! cos2x
2
Power Reduction Formulas
1+ cos2x
cos2 x =
2
Notice that the Power Reduction Formulas are just another arrangement of the
Double Angle Identities from trig: cos 2 x = 1 ! 2 sin2 x and cos 2 x = 2 cos2 x ! 1
sin2 x =
Strategies
1.
When the power of sin x or cos x is odd and positive, keep one factor of sin x
and convert all other factors to cos x, OR keep one factor of cos x and convert all
other factors to sin x. Use sin x dx or cos x dx as your du for u-substitution.
Example:
3
4
! sin x cos x dx
2.
When the powers of both sine and/or cosine are even (you can't use the first
strategy) make repeated use of the power reducing identities above.
Example:
4
! cos x dx
In this section we will be solving integrals of the form:
m
n
! sec x tan x dx
Strategies:
1.
If the power of secant x is even and positive, save a secant squared factor for
the du and convert the remaining factors to tangent x. Use the identity:
tan 2 x + 1 = sec 2 x
Example:
4
3
! sec x tan x dx
2.
If the power of tangent is odd and positive, save a secant-tangent factor for
the du, and convert the remaining factors to secants.
Example:
3
! tan x sec x dx
3.
If there are no secant factors and the power of tangent is even and positive,
convert a tangent-squared factor to a secant-squared factor. Let the secant-squared
factor be your du.
Example:
4
! tan x dx
4.
If there are no tangent factors and the power of secant is odd and positive,
use Integration by parts.
Example:
5.
3
! sec x dx
If none of the 4 strategies above apply, try converting to sines and cosines!
Example:
!
sec x
dx
tan 2 x