表格

• ${\displaystyle \int k\mathrm{d}x=kx+C(k\in \mathbb{R})}$
• ${\displaystyle \int x^{\alpha }\mathrm{d}x={\displaystyle \frac{x^{\alpha +1}}{\alpha +1}}+C(\alpha \ne -1)}$
  • ${\displaystyle \int {\displaystyle \frac{\mathrm{d}x}{x}}=\ln |x|+C}$
    • 注意绝对值！
• ${\displaystyle \int a^x\mathrm{d}x={\displaystyle \frac{a^x}{\ln a}}+C(a>0,a\ne 1)}$
  • ${\displaystyle \int e^x\mathrm{d}x=e^x+C}$
• ${\displaystyle \int \sin x\mathrm{d}x=-\cos x+C}$
• ${\displaystyle \int \cos x\mathrm{d}x=\sin x+C}$
• ${\displaystyle \int {\sec }^{2}x\mathrm{d}x=\tan x+C}$
• ${\displaystyle \int {\csc }^{2}x\mathrm{d}x=-\cot x+C}$
• ${\displaystyle \int \sec x\tan x\mathrm{d}x=\sec x+C}$
• ${\displaystyle \int \csc x\cot x\mathrm{d}x=-\csc x+C}$

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• ${\displaystyle \int {\displaystyle \frac{\mathrm{d}x}{\sqrt{a^2-x^2}}}=\arcsin {\displaystyle \frac{x}{a}}+C(a>0)}$
• ${\displaystyle \int {\displaystyle \frac{\mathrm{d}x}{x^2+a^2}}={\displaystyle \frac{1}{a}}\arctan {\displaystyle \frac{x}{a}}+C(a\ne 0)}$
• ${\displaystyle \int {\displaystyle \frac{\mathrm{d}x}{x^2-a^2}}={\displaystyle \frac{1}{2a}}\ln \left|{\displaystyle \frac{x-a}{x+a}}\right|+C}$
• ${\displaystyle \int \sec x\mathrm{d}x=\ln |\sec x+\tan x|+C}$
• ${\displaystyle \int \csc x\mathrm{d}x=\ln |\csc x-\cot x|+C=\ln |\tan {\displaystyle \frac{x}{2}}|+C}$
• ${\displaystyle \int {\displaystyle \frac{\mathrm{d}x}{\sqrt{x^2+a^2}}}=\ln (x+\sqrt{x^2+a^2})+C(a>0)}$
• ${\displaystyle \int {\displaystyle \frac{\mathrm{d}x}{\sqrt{x^2-a^2}}}=\ln (x+\sqrt{x^2-a^2})+C(a>0)}$

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• ${\displaystyle \int \sqrt{x^2-a^2}\mathrm{d}x={\displaystyle \frac{x}{2}}\sqrt{x^2-a^2}-{\displaystyle \frac{a^2}{2}}\ln |x+\sqrt{x^2-a^2}|+C}$
• ${\displaystyle \int \sqrt{x^2+a^2}\mathrm{d}x={\displaystyle \frac{x}{2}}\sqrt{x^2+a^2}+{\displaystyle \frac{a^2}{2}}\ln |x+\sqrt{x^2+a^2}|+C}$
• ${\displaystyle \int \sqrt{a^2-x^2}\mathrm{d}x={\displaystyle \frac{x}{2}}\sqrt{a^2-x^2}+{\displaystyle \frac{a^2}{2}}\arcsin {\displaystyle \frac{x}{a}}+C}$

部分公式的证明

• ${\displaystyle \int \csc x\mathrm{d}x=\ln |\csc x-\cot x|+C=\ln |\tan {\displaystyle \frac{x}{2}}|+C}$
  • $$\begin{array}{rl}\int \csc x& =\int \frac{\sin x}{{\sin }^{2}x}\mathrm{d}x\\ & =\int \frac{-1}{1-{\cos }^{2}x}\mathrm{d}\cos x\\ & =\int \frac{1}{{\cos }^{2}x-1}\mathrm{d}\cos x\\ & =\frac{1}{2}\ln \left|\frac{\cos x-1}{\cos x+1}\right|+C\\ & =\frac{1}{2}\ln \left|\frac{(\cos x-1{)}^2}{(\cos x+1)(\cos x-1)}\right|+C\\ & =\frac{1}{2}\ln \left|\frac{(\cos x-1{)}^2}{{\sin }^{2}x}\right|+C\\ & =\ln \left|\frac{\cos x-1}{\sin x}\right|+C\\ & =\ln |\csc x-\cot x|+C\end{array}$$
  • 万能代换 令 $u=\tan {\displaystyle \frac{x}{2}}⟹\mathrm{d}x={\displaystyle \frac{2\mathrm{d}u}{1+u^2}}$$$\begin{array}{rl}\int \csc xv& =\int \frac{{\displaystyle \frac{2}{1+u^2}}}{{\displaystyle \frac{2u}{1+u^2}}}\\ & =\int \frac{1}{u}\mathrm{d}u\\ & =\ln |u|+C\\ & =\ln |\tan \frac{x}{2}|\end{array}$$
• ${\displaystyle \int \sqrt{x^2-a^2}\mathrm{d}x={\displaystyle \frac{x}{2}}\sqrt{x^2-a^2}-{\displaystyle \frac{a^2}{2}}\ln |x+\sqrt{x^2-a^2}|+C}$
  • $$\begin{array}{rl}\int \sqrt{x^2-1}\mathrm{d}x& =\int \frac{x^2-1}{\sqrt{x^2-1}}\mathrm{d}x\\ & =\int \frac{x^2}{\sqrt{x^2-1}}\mathrm{d}x-\int \frac{1}{\sqrt{x^2-1}}\mathrm{d}x\\ & =\int x\mathrm{d}\sqrt{x^2-1}-\ln |\sqrt{x^2-1}+x|\\ & =x\sqrt{x^2-1}-\int \sqrt{x^2-1}\mathrm{d}x-\ln |\sqrt{x^2-1}+x|\end{array}$$