Integrate cos (lnx) dx Integrating by parts:
I'm having a hard time choosing the $u$, $du$, $v$ and $dv$...
I gave it a shot.
$u = \ln x \implies du = 1/x \ dx$
$v= \  ?$
$dv =  \cos \ dx$

I think you're better off using $u$-substitution here. Setting $\ln x = u$, you get $du = \frac{dx}{x} = \frac{dx}{e^u}$, and so, $dx = e^u\,du$. Then, your integral becomes
$$\int \cos\left( \ln x \right) \, dx = \int e^u\cos u \, du,$$
which you can evaluate by using integration by parts twice.
