TITLE: How to solve $\int(\cos(x)^{\cos(x)+1}\tan(x) (1+\log(\cos(x)))dx$? (2017 MIT Integration Bee Qualifier Problem #20)
QUESTION [5 upvotes]: I'm working my way through the qualifier, trying to learn some new techniques as I go. I am pretty stymied by this one. Solve: $$\int(\cos(x)^{\cos(x)+1}\tan(x)
(1+\log(\cos(x)))dx$$ My steps so far have been to let $u=\cos(x)$ and $\,du=-\sin(x)\,dx$ which yields: $$-\int u^u(1+\log(u))du$$ From here I am not so sure where to go. I would prefer a hint rather than a full solution, please. Thanks!

REPLY [8 votes]: That's all you need. Use $u^u=e^{u\log u}$ and differentiate.

REPLY [6 votes]: It's hard to hint this without giving it away, but what happens when you differentiate $x^x$?