How to integrate tan^3(x)sec(x)?

To integrate tan^3(x)sec(x), use substitution and integration by parts.

First, substitute u = tan(x) and du = sec^2(x) dx. This gives:

∫ tan^3(x)sec(x) dx = ∫ u^3 du

Next, integrate u^3 using the power rule:

∫ u^3 du = (1/4)u^4 + C

Substituting back in for u, we get:

∫ tan^3(x)sec(x) dx = (1/4)tan^4(x) + C

However, this is not the final answer. We need to check for any potential issues with the substitution. In this case, we need to make sure that u = tan(x) is a valid substitution for the entire domain of the original function.

Since tan(x) is undefined at x = (n + 1/2)π for any integer n, we need to exclude these values from the domain of the original function. Therefore, the final answer is:

∫ tan^3(x)sec(x) dx = (1/4)tan^4(x) + C, x ≠ (n + 1/2)π