We learn various techniques for integrating certain combinations of trigonometric functions.
(Video) Calculus: Single Variable
Online Texts
- OpenStax II 3.2: Trigonometric Integrals
- Ximera OSU: Trigonometric Integrals
- Community Calculus 8.2: Powers of Sine and Cosine
Examples
- When dealing with products of sines and cosines of the same quantity (in this
case, the is the same inside both the sine and the cosine), we can
use a substitution. We look for one with an odd power and build a
substitution using the other. In this case, we would use the substitution in the first integral andin the second.
- Using the substitutions just identified, we have in the former case and in the
latter. This means the integrals become
- We continue to simplify, using the trig identity to completely eliminate all
references to the variable in the integrand:
- Now we calculate the integral, giving in the first case andin the second.
- To conclude, we reverse the substitution, so that and
- When both powers are even, your only option is to use a trigonometric
identity to reduce the power. In this case, the identities are
- This means that
- Because we still have only even powers, we should use power reduction
again:
- Integrating this last expression gives
- In this case we look for either an even power of , which indicates a tangent
substitution, or an odd number of both secant and tangent, which indicates
a secant substitution. So we substitute in the first integral and in the
second. To simplify and write the integrands in terms of only, use the
identity . This gives for the two integrals (don’t forget the extra factor of coming from the chain rule).
- Integrating and reversing the substitution gives and