definite integral

1. \int_{0}^{\pi}{\frac{\sin xdx}{7+4\cos2x+\cos^22x}}

=\frac{1}{4}\int_{-1}^{1}{\frac{dt}{t^4+t^{2}+1}}=\frac{1}{2}\int_{0}^{1}{\frac{dt}{t^4+t^{2}+1}}

\text{Now, } \frac{1}{t^4+t^2+1}=\frac{At+B}{t^2+t+1}+\frac{Ct+D}{t^2-t+1}

\text{Which gives } A=\frac{1}{2},B=\frac{1}{2},C=-\frac{1}{2},D=\frac{1}{2}

Now, the integral reduces to well-known forms and evalutes to \boxed{\frac{\pi\sqrt{3}+3\ln3}{24}} .

2. \: I=\int_{0}^{\pi}{\frac{x\sin xdx}{7+4\cos2x+\cos^22x}}

\text{Also, }I=\int_{0}^{\pi}{\frac{\left( \pi-x\right)\sin xdx}{7+4\cos2x+\cos^22x}}

\therefore \: 2I= \pi\int_{0}^{\pi}{\frac{\sin xdx}{7+4\cos2x+\cos^22x}}

\Rightarrow I=\frac{\pi}{2}\int_{0}^{\pi}{\frac{\sin xdx}{7+4\cos2x+\cos^22x}}=\boxed{\frac{\pi}{2}\left(\frac{\pi\sqrt{3}+3\ln3}{24} \right)}