Question

How do you integrate `int xsec^2x` by integration by parts method?

Answer 1

`xtanx + ln|cosx|+C`

Explanation:

`int xsec^2x dx`

Integration by parts: `int f(x)g'(x) dx = f(x)g(x) - int f'(x)g(x) dx`

In this example: `f(x)=x -> f'(x) = 1`
and `g'(x)=sec^2x -> g(x) = tanx`

Therefore: `int xsec^2x dx = xtanx - int(1*tanx) dx`

`int xsec^2x dx = xtanx - int tanx dx`

`= xtanx + ln|cosx|+C` (Standard integral)