试计算 $dps{int_0^{cfrac{pi}{2}}cfrac{x^2}{sin^2x} d x}$.
解答: $$eex ea int_0^{cfrac{pi}{2}}cfrac{x^2}{sin^2x} d x &=-int_0^{cfrac{pi}{2}} x^2 d cot x\ &=2int_0^{cfrac{pi}{2}} x cot x d x\ &=2int_0^{cfrac{pi}{2}} x d ln sin x\ &=-2int_0^{cfrac{pi}{2}} ln sin x d x. eea eeex$$ 往求 $$eex ea int_0^{cfrac{pi}{2}} ln sin x d x &=int_0^{cfrac{pi}{2}} ln cos x d xquadsex{cfrac{pi}{2}-xleftrightsquigarrow x}\ &=cfrac{1}{2}int_0^{cfrac{pi}{2}} ln sin x+ln cos x d x\ &=cfrac{1}{2}int_0^{cfrac{pi}{2}} ln sin 2x d x-cfrac{pi}{4}ln 2\ &=cfrac{1}{4}int_0^{cfrac{pi}{2}} ln sin 2x d x-cfrac{pi}{4}ln 2quadsex{2xleftrightsquigarrow x}\ &=cfrac{1}{4}sez{ int_0^{cfrac{pi}{2}} ln sin x d x+int_0^{cfrac{pi}{2}} ln cos x d x }-cfrac{pi}{4}ln 2\&quadsex{x-cfrac{pi}{2}leftrightsquigarrow x}\ &=cfrac{1}{2}int_0^{cfrac{pi}{2}} ln sin x-cfrac{pi}{4}ln 2. eea eeex$$ 于是 $$ex int_0^{cfrac{pi}{2}} lnsin x d x=-cfrac{pi}{2}ln 2,quad int_0^{cfrac{pi}{2}} cfrac{x^2}{sin^2x} d x=pi ln 2. eex$$