3:45 PM
Solution in this answer is, in my opinion, very clever. And the story mentioned in the comment makes it even more interesting.
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Let $I:=\int\frac{\cos x}{\cos x+\sin x}dx$ and $J:=\int\frac{\sin x}{\cos x+\sin x}dx$. Then $I+J=x+C$, and $$I-J=\int\frac{\cos x-\sin x}{\cos x+\sin x}dx=\int\frac{u'(x)}{u(x)}dx,$$ where $u(x)=\cos x+\sin x$. Now we can conclude.

In this math education article the author describes giving the same problem to a young Terence Tao, aged 8; he gave essentially the same beautiful solution. — Erick Wong Aug 9 '12 at 18:29

Hello