就是灵活运用这么个公式： $$ \int_{a}^{b} f(x) , dx = \int_{a}^{b} f(a + b - x) , dx $$
具体形式
- $\int_{0}^{\pi} xf(\sin x) , \mathrm{d}x=\frac{\pi}{2}\int_{0}^{\pi} f(\sin x) , \mathrm{d}x$
- $\int_{1}^{a} f\left( x^{2}+\frac{a^{2}}{x^{2}} \right) , \frac{\mathrm{d}x}{x}=\int_{1}^{a} f(x+\frac{a^{2}}{x}) , \frac{\mathrm{d}x}{x}$ 换元 $t=x^{2}$ 和 $u=\frac{a^{2}}{t}$
- $\int_{0}^{2\pi} f(a\cos x+b\sin x) , \mathrm{d}x=2\int_{0}^{\pi} f(\sqrt{ a^{2}+b^{2} }\cos x) , \mathrm{d}x$ 辅助角+区间平移