Evaluate: ∫_(π/4)^(3π/4) x/(1+sin x) dx (Evaluate)

Let $I = \int_{\pi/4}^{3\pi/4}\frac{x}{1+\sin x}dx$. Using $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$ with $a+b = \pi$: $I = \int_{\pi/4}^{3\pi/4}\frac{\pi - x}{1+\sin x}dx$. Adding the two, $2I = \pi\int_{\pi/4}^{3\pi/4}\frac{dx}{1+\sin x} = \pi\int_{\pi/4}^{3\pi/4}(\sec^2x - \sec x\tan x)\,dx = \pi[\tan x - \sec x]_{\pi/4}^{3\pi/4}$. Now $\tan\frac{3\pi}{4} - \sec\frac{3\pi}{4} = -1 + \sqrt2$ and $\tan\frac{\pi}{4} - \sec\frac{\pi}{4} = 1 - \sqrt2$, so $2I = 2\pi(\sqrt2 - 1)$. Ans: $I = (\sqrt{2}-1)\pi$