跳转至

积分表

本文是高等数学（同济大学）附录的积分表。

含有 \(ax+b\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{a x+b}=\frac{1}{a} \ln |a x+b|+C\)

  2. \(\displaystyle \int(a x+b)^{\mu} \mathrm{d} x=\frac{1}{a(\mu+1)}(a x+b)^{\mu + 1}+C(\mu \neq-1)\)

  3. \(\displaystyle \int \frac{x}{a x+b} \mathrm{~d} x=\frac{1}{a^{2}}(a x+b-b \ln |a x+b|)+C\)

  4. \(\displaystyle \int \frac{x^{2}}{a x+b} \mathrm{~d} x=\frac{1}{a^{3}}\left[\frac{1}{2}(a x+b)^{2}-2 b(a x+b)+b^{2} \ln |a x+b|\right]+C\)

  5. \(\displaystyle \int \frac{\mathrm{d} x}{x(a x+b)}=-\frac{1}{b} \ln \left|\frac{a x+b}{x}\right|+C\)

  6. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2}(a x+b)}=-\frac{1}{b x}+\frac{a}{b^{2}} \ln \left|\frac{a x+b}{x}\right|+C\)

  7. \(\displaystyle \int \frac{x}{(a x+b)^{2}} \mathrm{~d} x=\frac{1}{a^{2}}\left(\ln |a x+b|+\frac{b}{a x+b}\right)+C\)

  8. \(\displaystyle \int \frac{x^{2}}{(a x+b)^{2}} \mathrm{~d} x=\frac{1}{a^{3}}\left(a x+b-2 b \ln |a x+b|-\frac{b^{2}}{a x+b}\right)+C\)

  9. \(\displaystyle \int \frac{\mathrm{d} x}{x(a x+b)^{2}}=\frac{1}{b(a x+b)}-\frac{1}{b^{2}} \ln \left|\frac{a x+b}{x}\right|+C\)

含有 \(\sqrt{a x+b}\) 的积分

  1. \(\displaystyle \int \sqrt{a x+b} d x=\frac{2}{3 a} \sqrt{(a x+b)^{3}}+C\)

  2. \(\displaystyle \int x \sqrt{a x+b} \mathrm{~d} x=\frac{2}{15 a^{2}}(3 a x-2 b) \sqrt{(a x+b)^{3}}+C\)

  3. \(\displaystyle \int x^{2} \sqrt{a x+b} \mathrm{~d} x=\frac{2}{105 a^{3}}\left(15 a^{2} x^{2}-12 a b x+8 b^{2}\right) \sqrt{(a x+b)^{3}}+C\)

  4. \(\displaystyle \int \frac{x}{\sqrt{a x+b}} \mathrm{~d} x=\frac{2}{3 a^{2}}(a x-2 b) \sqrt{a x+b}+C\)

  5. \(\displaystyle \int \frac{x^{2}}{\sqrt{a x+b}} \mathrm{~d} x=\frac{2}{15 a^{3}}\left(3 a^{2} x^{2}-4 a b x+8 b^{2}\right) \sqrt{a x+b}+C\)

  6. \(\displaystyle \int \frac{\mathrm{d} x}{x \sqrt{a x+b}}=\left\{\begin{array}{ll}\displaystyle \frac{1}{\sqrt{b}} \ln \left|\frac{\sqrt{a x+b}-\sqrt{b}}{\sqrt{a x+b}+\sqrt{b}}\right|+C & (b>0) \\\displaystyle \frac{2}{\sqrt{-b}} \arctan \sqrt{\frac{a x+b}{-b}}+C & (b<0)\end{array}\right.\)

  7. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2} \sqrt{a x+b}}=-\frac{\sqrt{a x+b}}{b x}-\frac{a}{2 b} \int \frac{\mathrm{d} x}{x \sqrt{a x+b}}\)

  8. \(\displaystyle \int \frac{\sqrt{a x+b}}{x} \mathrm{~d} x=2 \sqrt{a x+b}+b \int \frac{\mathrm{d} x}{x \sqrt{a x+b}}\)

  9. \(\displaystyle \int \frac{\sqrt{a x+b}}{x^{2}} \mathrm{~d} x=-\frac{\sqrt{a x+b}}{x}+\frac{a}{2} \int \frac{\mathrm{d} x}{x \sqrt{a x+b}}\)

含有 \(x^{2} \pm a^{2}\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2}+a^{2}}=\frac{1}{a} \arctan \frac{x}{a}+C\)

  2. \(\displaystyle \int \frac{\mathrm{d} x}{\left(x^{2}+a^{2}\right)^{n}}=\frac{x}{2(n-1) a^{2}\left(x^{2}+a^{2}\right)^{n-1}}+\frac{2 n-3}{2(n-1) a^{2}} \int \frac{\mathrm{d} x}{\left(x^{2}+a^{2}\right)^{n-1}}\)

  3. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2}-a^{2}}=\frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+C\)

含有 \(a x^{2}+b(a>0)\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{a x^{2}+b}=\left\{\begin{array}{l}\displaystyle \frac{1}{\sqrt{a b}} \arctan \sqrt{\frac{a}{b}} x+C \quad(b>0) \\\displaystyle \frac{1}{2 \sqrt{-a b}} \ln \left|\frac{\sqrt{a} x-\sqrt{-b}}{\sqrt{a} x+\sqrt{-b}}\right|+C \quad(b<0)\end{array}\right.\)

  2. \(\displaystyle \int \frac{x}{a x^{2}+b} \mathrm{~d} x=\frac{1}{2 a} \ln \left|a x^{2}+b\right|+C\)

  3. \(\displaystyle \int \frac{x^{2}}{a x^{2}+b} \mathrm{~d} x=\frac{x}{a}-\frac{b}{a} \int \frac{\mathrm{d} x}{a x^{2}+b}\)

  4. \(\displaystyle \int \frac{\mathrm{d} x}{x\left(a x^{2}+b\right)}=\frac{1}{2 b} \ln \frac{x^{2}}{\left|a x^{2}+b\right|}+C\)

  5. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2}\left(a x^{2}+b\right)}=-\frac{1}{b x}-\frac{a}{b} \int \frac{\mathrm{d} x}{a x^{2}+b}\)

  6. \(\displaystyle \int \frac{\mathrm{d} x}{x^{3}\left(a x^{2}+b\right)}=\frac{a}{2 b^{2}} \ln \frac{\left|a x^{2}+b\right|}{x^{2}}-\frac{1}{2 b x^{2}}+C\)

  7. \(\displaystyle \int \frac{\mathrm{d} x}{\left(a x^{2}+b\right)^{2}}=\frac{x}{2 b\left(a x^{2}+b\right)}+\frac{1}{2 b} \int \frac{\mathrm{d} x}{a x^{2}+b}\)

含有 \(a x^{2}+b x+c \quad(a>0)\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{a x^{2}+b x+c}=\left\{\begin{array}{l}\displaystyle \frac{2}{\sqrt{4 a c-b^{2}}} \arctan \frac{2 a x+b}{\sqrt{4 a c-b^{2}}}+C \quad\left(b^{2}<4 a c\right) \\\displaystyle \frac{1}{\sqrt{b^{2}-4 a c}} \ln \left|\frac{2 a x+b-\sqrt{b^{2}-4 a c}}{2 a x+b+\sqrt{b^{2}-4 a c}}\right|+C \quad\left(b^{2}>4 a c\right)\end{array}\right.\)

  2. \(\displaystyle \int \frac{x}{a x^{2}+b x+c} \mathrm{~d} x=\frac{1}{2 a} \ln \left|a x^{2}+b x+c\right|-\frac{b}{2 a} \int \frac{\mathrm{d} x}{a x^{2}+b x+c}\)

含有 \(\sqrt{x^{2}+a^{2}}(a>0)\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{x^{2}+a^{2}}}=\operatorname{arsh} \frac{x}{a}+C_{1}=\ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

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

  3. \(\displaystyle \int \frac{x}{\sqrt{x^{2}+a^{2}}} \mathrm{~d} x=\sqrt{x^{2}+a^{2}}+C\)

  4. \(\displaystyle \int \frac{x}{\sqrt{\left(x^{2}+a^{2}\right)^{3}}} \mathrm{~d} x=-\frac{1}{\sqrt{x^{2}+a^{2}}}+C\)

  5. \(\displaystyle \int \frac{x^{2}}{\sqrt{x^{2}+a^{2}}} \mathrm{~d} x=\frac{x}{2} \sqrt{x^{2}+a^{2}}-\frac{a^{2}}{2} \ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

  6. \(\displaystyle \int \frac{x^{2}}{\sqrt{\left(x^{2}+a^{2}\right)^{3}}} \mathrm{~d} x=-\frac{x}{\sqrt{x^{2}+a^{2}}}+\ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

  7. \(\displaystyle \int \frac{\mathrm{d} x}{x \sqrt{x^{2}+a^{2}}}=\frac{1}{a} \ln \frac{\sqrt{x^{2}+a^{2}}-a}{|x|}+C\)

  8. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2} \sqrt{x^{2}+a^{2}}}=-\frac{\sqrt{x^{2}+a^{2}}}{a^{2} x}+C\)

  9. \(\displaystyle \int \sqrt{x^{2}+a^{2}} \mathrm{~d} x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

  10. \(\displaystyle \int \sqrt{\left(x^{2}+a^{2}\right)^{3}} \mathrm{~d} x=\frac{x}{8}\left(2 x^{2}+5 a^{2}\right) \sqrt{x^{2}+a^{2}}+\frac{3}{8} a^{4} \ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

  11. \(\displaystyle \int x \sqrt{x^{2}+a^{2}} \mathrm{~d} x=\frac{1}{3} \sqrt{\left(x^{2}+a^{2}\right)^{3}}+C\)

  12. \(\displaystyle \int x^{2} \sqrt{x^{2}+a^{2}} \mathrm{~d} x=\frac{x}{8}\left(2 x^{2}+a^{2}\right) \sqrt{x^{2}+a^{2}}-\frac{a^{4}}{8} \ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

  13. \(\displaystyle \int \frac{\sqrt{x^{2}+a^{2}}}{x} \mathrm{~d} x=\sqrt{x^{2}+a^{2}}+a \ln \frac{\sqrt{x^{2}+a^{2}}-a}{|x|}+C\)

  14. \(\displaystyle \int \frac{\sqrt{x^{2}+a^{2}}}{x^{2}} \mathrm{~d} x=-\frac{\sqrt{x^{2}+a^{2}}}{x}+\ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

含有 \(\sqrt{x^{2}-a^{2}}(a>0)\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{x^{2}-a^{2}}}=\frac{x}{|x|} \operatorname{arch} \frac{|x|}{a}+C_{1}=\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\)

  2. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}}=-\frac{x}{a^{2} \sqrt{x^{2}-a^{2}}}+C\)

  3. \(\displaystyle \int \frac{x}{\sqrt{x^{2}-a^{2}}} \mathrm{~d} x=\sqrt{x^{2}-a^{2}}+C\)

  4. \(\displaystyle \int \frac{x}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}} \mathrm{~d} x=-\frac{1}{\sqrt{x^{2}-a^{2}}}+C\)

  5. \(\displaystyle \int \frac{x^{2}}{\sqrt{x^{2}-a^{2}}} \mathrm{~d} x=\frac{x}{2} \sqrt{x^{2}-a^{2}}+\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\)

  6. \(\displaystyle \int \frac{x^{2}}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}} \mathrm{~d} x=-\frac{x}{\sqrt{x^{2}-a^{2}}}+\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\)

  7. \(\displaystyle \int \frac{\mathrm{d} x}{x \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \arccos \frac{a}{|x|}+C\)

  8. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2} \sqrt{x^{2}-a^{2}}}=\frac{\sqrt{x^{2}-a^{2}}}{a^{2} x}+C\)

  9. \(\displaystyle \int \sqrt{x^{2}-a^{2}} \mathrm{~d} x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\)

  10. \(\displaystyle \int \sqrt{\left(x^{2}-a^{2}\right)^{3}} \mathrm{~d} x=\frac{x}{8}\left(2 x^{2}-5 a^{2}\right) \sqrt{x^{2}-a^{2}}+\frac{3}{8} a^{4} \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\)

  11. \(\displaystyle \int x \sqrt{x^{2}-a^{2}} \mathrm{~d} x=\frac{1}{3} \sqrt{\left(x^{2}-a^{2}\right)^{3}}+C\)

  12. \(\displaystyle \int x^{2} \sqrt{x^{2}-a^{2}} \mathrm{~d} x=\frac{x}{8}\left(2 x^{2}-a^{2}\right) \sqrt{x^{2}-a^{2}}-\frac{a^{4}}{8} \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\)

  13. \(\displaystyle \int \frac{\sqrt{x^{2}-a^{2}}}{x} \mathrm{~d} x=\sqrt{x^{2}-a^{2}}-a \arccos \frac{a}{|x|}+C\)

  14. \(\displaystyle \int \frac{\sqrt{x^{2}-a^{2}}}{x^{2}} \mathrm{~d} x=-\frac{\sqrt{x^{2}-a^{2}}}{x}+\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\)

含有 \(\sqrt{a^{2}-x^{2}}(a>0)\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{a^{2}-x^{2}}}=\arcsin \frac{x}{a}+C\)

  2. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}}=\frac{x}{a^{2} \sqrt{a^{2}-x^{2}}}+C\)

  3. \(\displaystyle \int \frac{x}{\sqrt{a^{2}-x^{2}}} \mathrm{~d} x=-\sqrt{a^{2}-x^{2}}+C\)

  4. \(\displaystyle \int \frac{x}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}} \mathrm{~d} x=\frac{1}{\sqrt{a^{2}-x^{2}}}+C\)

  5. \(\displaystyle \int \frac{x^{2}}{\sqrt{a^{2}-x^{2}}} \mathrm{~d} x=-\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \arcsin \frac{x}{a}+C\)

  6. \(\displaystyle \int \frac{x^{2}}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}} \mathrm{~d} x=\frac{x}{\sqrt{a^{2}-x^{2}}}-\arcsin \frac{x}{a}+C\)

  7. \(\displaystyle \int \frac{\mathrm{d} x}{x \sqrt{a^{2}-x^{2}}}=\frac{1}{a} \ln \frac{a-\sqrt{a^{2}-x^{2}}}{|x|}+C\)

  8. \(\displaystyle \int \frac{\mathrm{d} x}{x^{2} \sqrt{a^{2}-x^{2}}}=-\frac{\sqrt{a^{2}-x^{2}}}{a^{2} x}+C\)

  9. \(\displaystyle \int \sqrt{a^{2}-x^{2}} \mathrm{~d} x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \arcsin \frac{x}{a}+C\)

  10. \(\displaystyle \int \sqrt{\left(a^{2}-x^{2}\right)^{3}} \mathrm{~d} x=\frac{x}{8}\left(5 a^{2}-2 x^{2}\right) \sqrt{a^{2}-x^{2}}+\frac{3}{8} a^{4} \arcsin \frac{x}{a}+C\)

  11. \(\displaystyle \int x \sqrt{a^{2}-x^{2}} \mathrm{~d} x=-\frac{1}{3} \sqrt{\left(a^{2}-x^{2}\right)^{3}}+C\)

  12. \(\displaystyle \int x^{2} \sqrt{a^{2}-x^{2}} \mathrm{~d} x=\frac{x}{8}\left(2 x^{2}-a^{2}\right) \sqrt{a^{2}-x^{2}}+\frac{a^{4}}{8} \arcsin \frac{x}{a}+C\)

  13. \(\displaystyle \int \frac{\sqrt{a^{2}-x^{2}}}{x} \mathrm{~d} x=\sqrt{a^{2}-x^{2}}+a \ln \frac{a-\sqrt{a^{2}-x^{2}}}{|x|}+C\)

  14. \(\displaystyle \int \frac{\sqrt{a^{2}-x^{2}}}{x^{2}} \mathrm{~d} x=-\frac{\sqrt{a^{2}-x^{2}}}{x}-\arcsin \frac{x}{a}+C\)

含有 \(\sqrt{ \pm a x^{2}+b x+c}(a>0)\) 的积分

  1. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{a x^{2}+b x+c}}=\frac{1}{\sqrt{a}} \ln \left|2 a x+b+2 \sqrt{a} \sqrt{a x^{2}+b x+c}\right|+C\)

  2. \(\displaystyle \int \sqrt{a x^{2}+b x+c} \mathrm{~d} x= \frac{2 a x+b}{4 a} \sqrt{a x^{2}+b x+c}+\frac{4 a c-b^{2}}{8 \sqrt{a^{3}}} \ln \left|2 a x+b+2 \sqrt{a} \sqrt{a x^{2}+b x+c}\right|+C\)

  3. \(\displaystyle \int \frac{x}{\sqrt{a x^{2}+b x+c}} \mathrm{~d} x=\frac{1}{a} \sqrt{a x^{2}+b x+c}-\frac{b}{2 \sqrt{a^{3}}} \ln \left|2 a x+b+2 \sqrt{a} \sqrt{a x^{2}+b x+c}\right|+C\)

  4. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{c+b x-a x^{2}}}=-\frac{1}{\sqrt{a}} \arcsin \frac{2 a x-b}{\sqrt{b^{2}+4 a c}}+C\)

  5. \(\displaystyle \int \sqrt{c+b x-a x^{2}} \mathrm{~d} x= \frac{2 a x-b}{4 a} \sqrt{c+b x-a x^{2}}+\frac{b^{2}+4 a c}{8 \sqrt{a^{3}}} \arcsin \frac{2 a x-b}{\sqrt{b^{2}+4 a c}}+C\)

  6. \(\displaystyle \int \frac{x}{\sqrt{c+b x-a x^{2}}} \mathrm{~d} x=-\frac{1}{a} \sqrt{c+b x-a x^{2}}+\frac{b}{2 \sqrt{a^{3}}} \arcsin \frac{2 a x-b}{\sqrt{b^{2}+4 a c}}+C\)

含有 \(\sqrt{ \pm \frac{x-a}{x-b}}\) 或 \(\sqrt{(x-a)(b-x)}\) 的积分

  1. \(\displaystyle \int \sqrt{\frac{x-a}{x-b}} \mathrm{~d} x=(x-b) \sqrt{\frac{x-a}{x-b}}+(b-a) \ln (\sqrt{|x-a|}+\sqrt{|x-b|})+C\)

  2. \(\displaystyle \int \sqrt{\frac{x-a}{b-x}} \mathrm{~d} x=(x-b) \sqrt{\frac{x-a}{b-x}}+(b-a) \arcsin \sqrt{\frac{x-a}{b-a}}+C\)

  3. \(\displaystyle \int \frac{\mathrm{d} x}{\sqrt{(x-a)(b-x)}}=2 \arcsin \sqrt{\frac{x-a}{b-a}}+C(a<b)\)

  4. \(\displaystyle \int \sqrt{(x-a)(b-x)} \mathrm{d} x= \frac{2 x-a-b}{4} \sqrt{(x-a)(b-x)}+\frac{(b-a)^{2}}{4} \arcsin \sqrt{\frac{x-a}{b-a}}+C(a<b)\)

含有三角函数的积分

  1. \(\displaystyle \int \sin x \mathrm{~d} x=-\cos x+C\)

  2. \(\displaystyle \int \cos x \mathrm{~d} x=\sin x+C\)

  3. \(\displaystyle \int \tan x \mathrm{~d} x=-\ln |\cos x|+C\)

  4. \(\displaystyle \int \cot x \mathrm{~d} x=\ln |\sin x|+C\)

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

  6. \(\displaystyle \int \csc x \mathrm{~d} x=\ln \left|\tan \frac{x}{2}\right|+C=\ln |\csc x-\cot x|+C\)

  7. \(\displaystyle \int \sec ^{2} x \mathrm{~d} x=\tan x+C\)

  8. \(\displaystyle \int \csc ^{2} x \mathrm{~d} x=-\cot x+C\)

  9. \(\displaystyle \int \sec x \tan x \mathrm{~d} x=\sec x+C\)

  10. \(\displaystyle \int \csc x \cot x \mathrm{~d} x=-\csc x+C\)

  11. \(\displaystyle \int \sin ^{2} x \mathrm{~d} x=\frac{x}{2}-\frac{1}{4} \sin 2 x+C\)

  12. \(\displaystyle \int \cos ^{2} x \mathrm{~d} x=\frac{x}{2}+\frac{1}{4} \sin 2 x+C\)

  13. \(\displaystyle \int \sin ^{n} x \mathrm{~d} x=-\frac{1}{n} \sin ^{n-1} x \cos x+\frac{n-1}{n} \int \sin ^{n-2} x \mathrm{~d} x\)

  14. $\displaystyle \int \cos ^{n} x \mathrm{~d} x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x \mathrm{~d} x $

  15. \(\displaystyle \int \frac{\mathrm{d} x}{\sin ^{n} x}=-\frac{1}{n-1} \cdot \frac{\cos x}{\sin ^{n-1} x}+\frac{n-2}{n-1} \int \frac{\mathrm{d} x}{\sin ^{n-2} x}\)

  16. \(\displaystyle \int \frac{\mathrm{d} x}{\cos ^{n} x}=\frac{1}{n-1} \cdot \frac{\sin x}{\cos ^{n-1} x}+\frac{n-2}{n-1} \int \frac{\mathrm{d} x}{\cos ^{n-2} x}\)

  17. \(\displaystyle \begin{aligned}\int \cos ^{m} x \sin ^{n} x \mathrm{~d} x & =\frac{1}{m+n} \cos ^{m-1} x \sin ^{n+1} x+\frac{m-1}{m+n} \int \cos ^{m-2} x \sin ^{n} x \mathrm{~d} x \\& =-\frac{1}{m+n} \cos ^{m+1} x \sin ^{n-1} x+\frac{n-1}{m+n} \int \cos ^{m} x \sin ^{n-2} x \mathrm{~d} x\end{aligned}\)

  18. \(\displaystyle \int \sin a x \cos b x \mathrm{~d} x=-\frac{1}{2(a+b)} \cos (a+b) x-\frac{1}{2(a-b)} \cos (a-b) x+C\)

  19. \(\displaystyle \int \sin a x \sin b x \mathrm{~d} x=-\frac{1}{2(a+b)} \sin (a+b) x+\frac{1}{2(a-b)} \sin (a-b) x+C\)

  20. \(\displaystyle \int \cos a x \cos b x \mathrm{~d} x=\frac{1}{2(a+b)} \sin (a+b) x+\frac{1}{2(a-b)} \sin (a-b) x+C\)

  21. \(\displaystyle \int \frac{\mathrm{d} x}{a+b \sin x}=\frac{2}{\sqrt{a^{2}-b^{2}}} \arctan \frac{\displaystyle a \tan \frac{x}{2}+b}{\sqrt{a^{2}-b^{2}}}+C\left(a^{2}>b^{2}\right)\)

  22. \(\displaystyle \int \frac{\mathrm{d} x}{a+b \sin x}=\frac{1}{\sqrt{b^{2}-a^{2}}} \ln \left|\frac{\displaystyle a \tan \frac{x}{2}+b-\sqrt{b^{2}-a^{2}}}{\displaystyle a \tan \frac{x}{2}+b+\sqrt{b^{2}-a^{2}}}\right|+C\left(a^{2}<b^{2}\right)\)

  23. \(\displaystyle \int \frac{\mathrm{d} x}{a+b \cos x}=\frac{2}{a+b} \sqrt{\frac{a+b}{a-b}} \arctan \left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right)+C\left(a^{2}>b^{2}\right)\)

  24. \(\displaystyle \int \frac{\mathrm{d} x}{a+b \cos x}=\frac{1}{a+b} \sqrt{\frac{a+b}{b-a}} \ln \left|\frac{\displaystyle \tan \frac{x}{2}+\sqrt{\frac{a+b}{b-a}}}{\displaystyle \tan \frac{x}{2}-\sqrt{\frac{a+b}{b-a}}}\right|+C\left(a^{2}<b^{2}\right)\)

  25. \(\displaystyle \int \frac{\mathrm{d} x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}=\frac{1}{a b} \arctan \left(\frac{b}{a} \tan x\right)+C\)

  26. \(\displaystyle \int \frac{\mathrm{d} x}{a^{2} \cos ^{2} x-b^{2} \sin ^{2} x}=\frac{1}{2 a b} \ln \left|\frac{b \tan x+a}{b \tan x-a}\right|+C\)

  27. \(\displaystyle \int x \sin a x \mathrm{~d} x=\frac{1}{a^{2}} \sin a x-\frac{1}{a} x \cos a x+C\)

  28. \(\displaystyle \int x^{2} \sin a x \mathrm{~d} x=-\frac{1}{a} x^{2} \cos a x+\frac{2}{a^{2}} x \sin a x+\frac{2}{a^{3}} \cos a x+C\)

  29. \(\displaystyle \int x \cos a x \mathrm{~d} x=\frac{1}{a^{2}} \cos a x+\frac{1}{a} x \sin a x+C\)

  30. \(\displaystyle \int x^{2} \cos a x \mathrm{~d} x=\frac{1}{a} x^{2} \sin a x+\frac{2}{a^{2}} x \cos a x-\frac{2}{a^{3}} \sin a x+C\)

含有反三角函数的积分（其中 \(a>0\)）

  1. \(\displaystyle \int \arcsin \frac{x}{a} \mathrm{~d} x=x \arcsin \frac{x}{a}+\sqrt{a^{2}-x^{2}}+C\)

  2. \(\displaystyle \int x \arcsin \frac{x}{a} \mathrm{~d} x=\left(\frac{x^{2}}{2}-\frac{a^{2}}{4}\right) \arcsin \frac{x}{a}+\frac{x}{4} \sqrt{a^{2}-x^{2}}+C\)

  3. \(\displaystyle \int x^{2} \arcsin \frac{x}{a} \mathrm{~d} x=\frac{x^{3}}{3} \arcsin \frac{x}{a}+\frac{1}{9}\left(x^{2}+2 a^{2}\right) \sqrt{a^{2}-x^{2}}+C\)

  4. \(\displaystyle \int \arccos \frac{x}{a} \mathrm{~d} x=x \arccos \frac{x}{a}-\sqrt{a^{2}-x^{2}}+C\)

  5. \(\displaystyle \int x \arccos \frac{x}{a} \mathrm{~d} x=\left(\frac{x^{2}}{2}-\frac{a^{2}}{4}\right) \arccos \frac{x}{a}-\frac{x}{4} \sqrt{a^{2}-x^{2}}+C\)

  6. \(\displaystyle \int x^{2} \arccos \frac{x}{a} \mathrm{~d} x=\frac{x^{3}}{3} \arccos \frac{x}{a}-\frac{1}{9}\left(x^{2}+2 a^{2}\right) \sqrt{a^{2}-x^{2}}+C\)

  7. \(\displaystyle \int \arctan \frac{x}{a} \mathrm{~d} x=x \arctan \frac{x}{a}-\frac{a}{2} \ln \left(a^{2}+x^{2}\right)+C\)

  8. \(\displaystyle \int x \arctan \frac{x}{a} \mathrm{~d} x=\frac{1}{2}\left(a^{2}+x^{2}\right) \arctan \frac{x}{a}-\frac{a}{2} x+C\)

  9. \(\displaystyle \int x^{2} \arctan \frac{x}{a} \mathrm{~d} x=\frac{x^{3}}{3} \arctan \frac{x}{a}-\frac{a}{6} x^{2}+\frac{a^{3}}{6} \ln \left(a^{2}+x^{2}\right)+C\)

含有指数函数的积分

  1. \(\displaystyle \int a^{x} \mathrm{~d} x=\frac{1}{\ln a} a^{x}+C\)

  2. \(\displaystyle \int \mathrm{e}^{a x} \mathrm{~d} x=\frac{1}{a} \mathrm{e}^{a x}+C\)

  3. \(\displaystyle \int x \mathrm{e}^{a x} \mathrm{~d} x=\frac{1}{a^{2}}(a x-1) \mathrm{e}^{a x}+C\)

  4. \(\displaystyle \int x^{n} \mathrm{e}^{a x} \mathrm{~d} x=\frac{1}{a} x^{n} \mathrm{e}^{a x}-\frac{n}{a} \int x^{n-1} \mathrm{e}^{a x} \mathrm{~d} x\)

  5. \(\displaystyle \int x a^{x} \mathrm{~d} x=\frac{x}{\ln a} a^{x}-\frac{1}{(\ln a)^{2}} a^{x}+C\)

  6. \(\displaystyle \int x^{n} a^{x} \mathrm{~d} x=\frac{1}{\ln a} x^{n} a^{x}-\frac{n}{\ln a} \int x^{n-1} a^{x} \mathrm{~d} x\)

  7. \(\displaystyle \int \mathrm{e}^{a x} \sin b x \mathrm{~d} x=\frac{1}{a^{2}+b^{2}} \mathrm{e}^{a x}(a \sin b x-b \cos b x)+C\)

  8. \(\displaystyle \int \mathrm{e}^{a x} \cos b x \mathrm{~d} x=\frac{1}{a^{2}+b^{2}} \mathrm{e}^{a x}(b \sin b x+a \cos b x)+C\)

  9. \(\displaystyle \int \mathrm{e}^{a x} \sin ^{n} b x \mathrm{~d} x= \frac{1}{a^{2}+b^{2} n^{2}} \mathrm{e}^{a x} \sin ^{n-1} b x(a \sin b x-n b \cos b x)+\frac{n(n-1) b^{2}}{a^{2}+b^{2} n^{2}} \int \mathrm{e}^{a x} \sin ^{n-2} b x \mathrm{~d} x\)

  10. \(\displaystyle \int \mathrm{e}^{a x} \cos ^{n} b x \mathrm{~d} x= \frac{1}{a^{2}+b^{2} n^{2}} \mathrm{e}^{a x} \cos ^{n-1} b x(a \cos b x+n b \sin b x)+\frac{n(n-1) b^{2}}{a^{2}+b^{2} n^{2}} \int \mathrm{e}^{a x} \cos ^{n-2} b x \mathrm{~d} x\)

含有对数函数的积分

  1. \(\displaystyle \int \ln x \mathrm{~d} x=x \ln x-x+C\)

  2. \(\displaystyle \int \frac{\mathrm{d} x}{x \ln x}=\ln |\ln x|+C\)

  3. \(\displaystyle \int x^{n} \ln x \mathrm{~d} x=\frac{1}{n+1} x^{n+1}\left(\ln x-\frac{1}{n+1}\right)+C\)

  4. \(\displaystyle \int(\ln x)^{n} \mathrm{~d} x=x(\ln x)^{n}-n \int(\ln x)^{n-1} \mathrm{~d} x\)

  5. \(\displaystyle \int x^{m}(\ln x)^{n} \mathrm{d} x=\frac{1}{m+1} x^{m+1}(\ln x)^{n}-\frac{n}{m+1} \int x^{m}(\ln x)^{n-1} \mathrm{~d} x\)

含有双曲函数的积分

  1. \(\displaystyle \int \operatorname{sh} x \mathrm{~d} x=\operatorname{ch} x+C\)

  2. \(\displaystyle \int \operatorname{ch} x \mathrm{~d} x=\operatorname{sh} x+C\)

  3. \(\displaystyle \int \text { th } x \mathrm{~d} x=\ln \operatorname{ch} x+C\)

  4. \(\displaystyle \int \operatorname{sh}^{2} x \mathrm{~d} x=-\frac{x}{2}+\frac{1}{4} \operatorname{sh} 2 x+C\)

  5. \(\displaystyle \int \operatorname{ch}^{2} x \mathrm{~d} x=\frac{x}{2}+\frac{1}{4} \operatorname{sh} 2 x+C\)

定积分

  1. \(\displaystyle \int_{-\pi}^{\pi} \cos n x \mathrm{~d} x=\int_{-\pi}^{\pi} \sin n x \mathrm{~d} x=0\)

  2. \(\displaystyle \int_{-\pi}^{\pi} \cos m x \sin n x \mathrm{~d} x=0\)

  3. \(\displaystyle \int_{-\pi}^{\pi} \cos m x \cos n x \mathrm{~d} x=\left\{\begin{array}{l}0, m \neq n \\\pi, m=n\end{array}\right.\)

  4. \(\displaystyle \int_{-\pi}^{\pi} \sin m x \sin n x \mathrm{~d} x=\left\{\begin{array}{ll}0, & m \neq n \\\pi, & m=n\end{array}\right.\)

  5. \(\displaystyle \int_{0}^{\pi} \sin m x \sin n x \mathrm{~d} x=\int_{0}^{\pi} \cos m x \cos n x \mathrm{~d} x=\left\{\begin{array}{l}0, m \neq n \\\displaystyle \frac{\pi}{2}, m=n\end{array}\right.\)

  6. \(\displaystyle I_{n}=\int_{0}^{\frac{\pi}{2}} \sin ^{n} x \mathrm{~d} x=\int_{0}^{\frac{\pi}{2}} \cos ^{n} x \mathrm{~d} x\)

\(\begin{aligned}I_{n} & =\frac{n-1}{n} I_{n-2} =\left\{\begin{array}{ll}\displaystyle \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \cdots \frac{4}{5} \cdot \frac{2}{3}&(n \text { 为大于 } 1 \text { 的正奇数 }), I_{1}=1 \\\displaystyle \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}&(n \text { 为正偶数 }),\displaystyle I_{0}=\frac{\pi}{2}\end{array}\right.\end{aligned}\)

评论