yes,,you can use quotient rule...
I lied, we cannot use by parts on that particular integral. It cannot be evaluated in elementary functions: Wolfram Mathematica Online Integrator
It's defined as
So it would appear as if I have taken you down a path where you can integrate the integrand but not in terms of elementary functions. I'm sure an alternative approach will yield a result that can be fully explained in terms of elementary functions (i'll think of another way to start it off).
Calling...Simplependulum!!