Trig Integrals with tangent and secant raised to powers
For an integral involving a product of powers of tangent and secant functions, a useful approach is to break down the integral using trigonometric identities and substitution.
When dealing with a mix of tangent and secant raised to odd powers, a standard strategy is to reserve a factor of secant and tangent to work with the remaining terms. For example, in this case, you can factor out a secant-tangent pair, which allows you to use the identity that relates the derivative of secant to secant-tangent. This makes substitution straightforward.
Once you isolate the secant-tangent pair, you can rewrite the remaining secant terms using a trigonometric identity, such as secant squared being equal to one plus tangent squared. This substitution simplifies the integral, making it easier to handle.
After applying these steps, you integrate the resulting expression by substituting and simplifying, eventually solving for the final result. Using this method helps break down complex trigonometric integrals into manageable parts.