What is the integral with tan and secant? ?

Question

integral (from pi/6 to pi/4) of (1+tanx)^5 * (sec^2(x)) dx ? The directions say to integrate by parts where you substitute a u and a dv, so please use that method, or at least what the substitutions would be and I can figure out the rest. Thanks!

Answer 1

I am not sure you need to do this in parts since d(tan(x)) is sec^2(x) dx

let U = 1 + tan(x)

dU = sec^2(x) dx

[1 + tan(x)]^5 (sec^2(x)dx = U^5 dU

This can easily be integrated to U^6/6 = [1 + tan(x)]^6/6

And you just need to evaluate it.

Answer 2

Let u = 1 + tan(x).  You should end up with the integral of u^5 du.