Question

How do you integrate `int x sec 5 x dx` using integration by parts?

Answer 1

I don't believe you do.

Explanation:

We can start with `u=x` and `dv=sec(5x)`, so that

`du = dx` and `v = int sec(5x) dx = 1/5 ln(tan5x+sec5x)`

We get,

`int x sec(5x) dx = x/5 ln(tan5x+sec5x) - int ln(tan5x+sec5x) dx`

If you have some way of finding that integral, use it.

Wolfram Alpha gives the original integral as:

`int x sec(5 x) dx = 1/25 i (Li_2(-i e^(5 i x))-Li_2(i e^(5 i x)))+1/5 x (log(1-i e^(5 i x))-log(1+i e^(5 i x)))+C`

Where `Li_n` is the polylogarithmic function.

You can read more about the polylogarithmic function here:

http://mathworld.wolfram.com/Polylogarithm.html