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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!

Asaf Karagila
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Mitchell Jacky
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2 Answers2

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That's all you need. Use $u^u=e^{u\log u}$ and differentiate.

anon
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It's hard to hint this without giving it away, but what happens when you differentiate $x^x$?

B. Mehta
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