<s>[INST] Integration of $x\cos(x)/(5+2\cos^2 x)$ on the interval from $0$ to $2\pi$ 
Compute the integral
  $$\int_{0}^{2\pi}\frac{x\cos(x)}{5+2\cos^2(x)}dx$$

My Try: I substitute $$\cos(x)=u$$
but it did not help. Please help me to solve this.Thanks 
 [/INST] Using $\displaystyle\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx,$
$$I=\int_0^{2\pi}\frac{x\cos x}{5+2\cos^2x}dx=\int_0^{2\pi}\frac{(2\pi-x)\cos(2\pi-x)}{5+2\cos^2(2\pi-x)}\ dx=\int_0^{2\pi}\frac{(2\pi-x)\cos x}{5+2\cos^2 x}\ dx$$
$$2I=2\pi\int_0^{2\pi}\frac{\cos x}{5+2\cos^2x}dx$$
$$\implies I=\pi\int_0^{2\pi}\frac{\cos x}{7-2\sin^2x}dx$$
Set $\displaystyle\sin x=u$
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