一、区间再现的表达

$$\int_a^bf(x)dx = \int_a^bf(a+b-x)dx = \frac{1}{2}\int_a^b[f(x)+f(a+b-x)]dx$$

证明：

$$\int_a^bf(a+b-x)dx\overset{令t=a+b-x}{=}\int_b^af(t)(-1)dt=\int_a^bf(t)dt=\int_a^bf(x)dx$$

例题

$$\begin{aligned} &1.1\qquad \int_0^\pi \frac{\sin x}{\sin x+\cos x}dx \\ &2.1\qquad \int_0^{\frac{\pi}{2}}\frac{1}{1+(\tan x)^\alpha}dx \\ &2.2\qquad \int_0^{+\inf}\frac{1}{(1+x^2)(1+x^{\alpha})}dx \\ &3.1\qquad \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\arctan e^x)\cdot \sin^2xdx \\ &4.1\qquad \int_{-2}^{2}x\cdot\ln(1+e^x)dx \\ &5.1\qquad \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{\cos^2x}{x(\pi-2x)}dx \\ &6.0\qquad \int_0^{\frac{\pi}{4}}\frac{x}{\cos(\frac{\pi}{4}-x)\cdot\cos x}dx \\ &6.1\qquad \int\frac{1}{\sin(x+a)\cdot\sin(x+b)} \\ &6.2\qquad \int\frac{1}{\sin(x+3)\cdot\sin(x+5)} \\ &7.0\qquad \int_0^{\frac{\pi}{2}}\ln\sin xdx \\ &7.1\qquad\int_0^{\frac{\pi}{2}}\frac{x}{\tan x}dx \\ &7.2\qquad \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos x\ln \cos x}{1+\sin x+\cos x}dx \\ &8.0\qquad \int_0^1\frac{\ln(1+x)}{1+x^2}dx \\ &8.1\qquad \int_0^1\frac{\arctan x}{1+x}dx \\ &9.0\qquad \int_0^{n\pi}x\cdot |\sin x| dx \\ &9.1\qquad a_n=\int_0^{n\pi}x\cdot |\sin x| dx, 求\sum_{n=1}^{\inf}\frac{1}{a_n} \\ &10.0\qquad \int_0^1\frac{\arcsin\sqrt{x}}{\sqrt{x^2-x+1}}dx \\ &11.0\qquad \int_0^1x\cdot\arcsin 2\sqrt{x-x^2}dx \\ &12.0\qquad \int_0^1(1-x)^{100}\cdot xdx \\ &13.0\qquad \int_0^2x\cdot(x-1)\cdot(x-2)dx \\ &13.1\qquad \int_0^{2n}x\cdot(x-1)\cdot(x-2)\cdot\cdot\cdot\cdot(x-2n) dx \end{aligned} \\$$

例题解析

之后再补吧。
这是看凯哥的学习笔记。原视频可以参考：
https://www.bilibili.com/video/BV1ah411Y789