Decompose to Fourier and calculate integral:
$$\int \limits_{0}^{2\pi} \frac{x\cos{mx}}{1-a\cos{x}}dx, \qquad |a| < 1$$
I have tried to use the below equations, but have not succeeded yet.
$$\frac{1-a\cos{bx}}{1-2a\cos{bx}+a^2}=1+\sum_{n=1}^{\infty} a^n\cdot \cos{nbx},\qquad |a|<1;$$ $$\frac{a\sin{bx}}{1-2a\cos{bx}+a^2}=\sum_{n=1}^{\infty} a^n\cdot \sin{nbx}, \qquad |a|<1;$$ $$\frac{1-a^2}{1-2a\cos{bx}+a^2}=1+2\sum_{n=1}^{\infty} a^n\cdot \cos{nbx}, \qquad |a|<1;$$
I got $$\int \limits_{0}^{2\pi} \frac{x(1 - (1 + \sum \limits_{n=1}^{+\infty} a^n\cos{nmx})(1-2a\cos{mx} + a^2))}{a(1+\sum \limits_{n=1}^{+\infty} a^n\cos{nx})(1-2a\cos{x} + a^2)}dx$$