Integrating trig functions in fractions
For an integral involving a quotient of powers of tangent and secant functions, such as this one, a key idea is recognizing how the derivative of the tangent function relates to secant squared.
Since the denominator is secant squared, which is the derivative of tangent, this suggests that substitution can simplify the integral. In this case, you can substitute the tangent of x as your new variable because the derivative of tangent appears in the denominator.
Once you make the substitution, the integral becomes much simpler, reducing it to an expression that can be integrated easily. After finding the antiderivative, you'll substitute back in terms of the original variable. This method leverages the relationship between the derivative of tangent and secant squared to streamline the problem.