Problems

By making the substitution \(x=\pi-t\,\), show that \[ \! \int_0^\pi x\f(\sin x) \d x = \tfrac12 \pi \! \int_0^\pi \f(\sin x) \d x\,, \] where \(\f(\sin x)\) is a given function of \(\sin x\). Evaluate the following integrals:

1. \(\displaystyle \int_0^\pi \frac {x \sin x}{3+\sin^2 x}\,\d x\,\);
2. $\displaystyle \int_0^{2\pi} \frac {x \sin x}{3+\sin^2 x}\,\d x\,\(;
3. \)\displaystyle \int_{0}^{\pi} \frac {x \big\vert\sin 2x\big\vert}{3+\sin^2 x}\,\d x\,$.