常用Riemann积分公式

常用Riemann积分公式总结

$$\int{\frac{x}{\sqrt{ax+b}}\mathrm{d}x=\frac{2}{3a^2}\left( ax-2b \right) \sqrt{ax+b}} \\$$

$$\int{\frac{\mathrm{d}x}{x^2-a^2}=\frac{1}{2a}\ln \left| \frac{x-a}{x+a} \right|} \\$$

$$\int{\frac{\mathrm{d}x}{x^2+a^2}=\frac{1}{a}\mathrm{arc}\tan \frac{x}{a}} \\$$

$$\int{\frac{\mathrm{d}x}{ax^2+bx+c}}=\begin{cases} \frac{2}{\sqrt{-\Delta}}\mathrm{arc}\tan \frac{2ax+b}{\sqrt{-\Delta}}\text{，}\Delta <0\\ \frac{1}{\sqrt{\Delta}}\ln \left| \frac{2ax+b-\sqrt{\Delta}}{2ax+b+\sqrt{\Delta}} \right|\text{，}\Delta >0\\ \end{cases} \\$$

$$\int{\frac{\mathrm{d}x}{\sqrt{x^2+a^2}}=\ln \left| x+\sqrt{x^2+a^2} \right|} \\$$

$$\int{\frac{\mathrm{d}x}{\sqrt{x^2-a^2}}=\ln \left| x+\sqrt{x^2-a^2} \right|} \\$$

$$\text{上面两个式子可以归并为}$$

$$\int{\frac{\mathrm{d}x}{\sqrt{x^2+y}}=\ln \left| x+\sqrt{x^2+y} \right|}\left( \text{不论}y\text{的正负} \right) \\$$

$$\int{\frac{\mathrm{d}x}{\left( x^2+a^2 \right) ^{\frac{3}{2}}}=\frac{x}{a^2\sqrt{x^2+a^2}}} \\$$

$$\int{\frac{\mathrm{d}x}{\sqrt{a^2-x^2}}}=\mathrm{arc}\sin \frac{x}{a} \\$$

$$\int{\sqrt{x^2+a^2}}\mathrm{d}x=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln \left| x+\sqrt{x^2+a^2} \right| \\$$

$$\int{\sqrt{x^2-a^2}}\mathrm{d}x=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln \left| x+\sqrt{x^2-a^2} \right| \\$$

$$\text{上面两个式子可以归并为}$$

$$\int{\sqrt{x^2+y}}\mathrm{d}x=\frac{x}{2}\sqrt{x^2+y}+\frac{y}{2}\ln \left| x+\sqrt{x^2+y} \right|\left( \text{不论}y\text{的正负} \right) \\$$

$$\int{\sqrt{a^2-x^2}}\mathrm{d}x=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\mathrm{arc}\sin \frac{x}{a} \\$$

$$\int{a^x}\mathrm{d}x=\frac{a^x}{\ln a} \\$$

$$\int{\tan x\mathrm{d}x=-\ln \left| \cos x \right|} \\$$

$$\int{\cot x\mathrm{d}x=}\ln \left| \sin x \right| \\$$

$$\int{\sec x\mathrm{d}x=}\ln \left| \tan \left( \frac{x}{2}+\frac{\pi}{4} \right) \right|=\ln \left( \sec x+\tan x \right) \\$$

$$\int{\csc x\mathrm{d}x=}\ln \left| \tan \left( \frac{x}{2} \right) \right| \\$$

$$\int{\sec ^2x\mathrm{d}x=}\tan x \\$$

$$\int{\csc ^2x\mathrm{d}x=}-\cot x \\$$

$$I_{m,n} = \int_{0}^{\frac{\pi}{2}} \sin^m x \cos^n x \,\mathrm{d}x = \begin{cases} \displaystyle \frac{(m-1)!!\,(n-1)!!}{(m+n)!!} \cdot \frac{\pi}{2}, & \text{当} m,n \text{都为偶数}\\ \displaystyle \frac{(m-1)!!\,(n-1)!!}{(m+n)!!}, & \text{否则} \end{cases}$$

$$\text{万能替换}: \\ \mathrm{d}x=\frac{2}{1+t^2}\mathrm{d}t,\sin x=\frac{2t}{1+t^2},\cos x=\frac{1-t^2}{1+t^2},t=\tan \left( \frac{x}{2} \right)$$