Question

How do you integrate `cos(log(x)) dx`?

Answer 1

(I will assume `logx` is natural log. (If not, insert ln10 where needed.)

`int cos(log(x))dx = int xcos(log(x))*1/xdx`

Let `u=x` and `dv` is the rest of the integrand.
Now we can integrate `v = int cos(log(x))*1/xdx = sin(log(x))`
(Use substitution with `w=log(x)`)

Parts gives us:

`int cos(log(x))dx = xsin(log(x)) - int sin(log(x)) dx`

Do the same trick again to get

`int cos(log(x))dx = xsin(log(x)) + xcos(log(x)) - int cos(log(x)) dx`

So

`int cos(log(x))dx = 1/2(xsin(log(x)) + xcos(log(x)))`