The value of _0^2(x^8x)/(^8x+^8x),dx is equal to:

Question

The value of $\displaystyle\int_0^{2\pi}\frac{x\sin^8x}{\sin^8x+\cos^8x}\,dx$ is equal to:

Accepted Answer

Using $\displaystyle\int_0^{2a}f(x)dx=\int_0^af(x)dx+\int_0^af(2a-x)dx$ with $a=\pi$:
$$I=\int_0^\pi\left[\frac{x\sin^8x}{\sin^8x+\cos^8x}+\frac{(2\pi-x)\sin^8x}{\sin^8x+\cos^8x}\right]dx=\int_0^\pi\frac{2\pi\sin^8x}{\sin^8x+\cos^8x}dx$$

$$=2\pi\int_0^{\pi/2}\left[\frac{\sin^8x}{\sin^8x+\cos^8x}+\frac{\cos^8x}{\sin^8x+\cos^8x}\right]dx=2\pi\int_0^{\pi/2}1\,dx=2\pi\cdot\frac{\pi}2=\pi^2$$

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