I'm compiling a list of interesting definite integrals for an upcoming blog post, and I thought that the math SE community might have a few interesting problems to offer. I am especially interested in integrals that use "tricks" that are hard to spot at first or are particularly elegant ways of solving seemingly difficult problems.
An example of such an integral is the following: $$\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$$ $$=\int_0^{\pi/2} 1-\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$$ $$=\int_0^{\pi/2} 1-\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$$ $$=\frac{\pi}{2}-\int_0^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$$ $$x\to \frac{\pi}{2}-x$$ $$=\frac{\pi}{2}-\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$$ And so if one lets $I$ be the value of the integral, $$I=\frac{\pi}{2}-I$$ and $$I=\frac{\pi}{4}$$ This integral holds a very good example of what an "integral trick" is, because it uses the properties of integrals to evaluate a very difficult-looking definite integral without actually evaluating it.
I look forward to seeing what everyone has to say!
