How to evaluate integrals of tan^m(x) sec^n(x)

In this video, we investigate the integrals of tan^m(x)secⁿ(x) where m and n are positive integers. We discuss strategies on how to approach each of the following 4 cases:

1. tan^m(x)sec²(x) - this is the easiest case where m is any power. Use the substitution u = tan(x). We discuss the example: tan³(x)sec²(x).

2. tan^m(x)secⁿ(x) where m is any power and n is even. Save a sec²(x). Then use sec²(x) = tan²(x) + 1 to get everything in terms of tan(x), then use the substitution u = tan(x). We discuss the example: tan⁵(x)sec⁴(x).

3. tan^m(x)secⁿ(x) where m and n are odd. Save a sec(x)tan(x). Use tan²(x) = sec²(x) - 1 to get everything in terms of sec(x), then use the substitution u = sec(x). We discuss the example: tan⁵(x)sec³(x).

4. tan^m(x)secⁿ(x) where m is even and n is odd. Use tan²(x) = sec²(x) - 1 to get everything in terms of sec(x) and then the integral purely becomes an integral of various powers of sec(x) with respect to x. We discuss the example: tan²(x)sec(x).

The 4th case is the most difficult because it requires the integration of odd powers of secⁿ(x).

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