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积分表二（高等数学同济版中所有的积分公式）

news 2025/7/7 14:32:52

文章目录

- 含有 $\sqrt{\pm \frac{x-a}{x+a}}$ 或者 $\sqrt{(x-a)(b-x)}$ 的积分
- 含有三角函数函数的积分
- 含有反三角函数的积分 (其中 $a > 0$ )
- 含有指数函数的积分
- 含有对数函数的积分
- 含有双曲函数的积分
- 定积分

含有 $\sqrt{\pm \frac{x-a}{x+a}}$ 或者 $\sqrt{(x-a)(b-x)}$ 的积分

$\begin{equation} 79.\,\int\!\! \sqrt{\frac{x-a}{x-b}}dx=(x-b)\sqrt{\frac{x-a}{x-b}}+(b-a)\ln(\sqrt{\vert x-a\vert} + \sqrt{\vert x-b \vert }\,) +C \end{equation}$

$\begin{equation} 80.\,\int\!\! \sqrt{\frac{x-a}{x-b}}\,dx=(x-b)\sqrt{\frac{x-a}{x-b}}+(b-a)\arcsin\sqrt{\frac{x-a}{b-a}}+C \end{equation}$

$\begin{equation} 81.\,\int\!\! \frac{dx}{\sqrt{(x-a)(x-b)}}=2\arcsin\sqrt{\frac{x-a}{b-a}}+C \quad (a<b) \end{equation}$

$\begin{equation} 82.\,\int\!\! \sqrt{(x-a)(b-x)}\,dx=\frac{2x-a-b}{4}\sqrt{(x-a)(b-x)} +\frac{(b-a)^2}{4}\arcsin\sqrt{\frac{x-a}{b-a}}+C \quad (a<b) \end{equation}$

含有三角函数函数的积分

$\begin{equation} 83.\,\int\!\! \sin x dx = -\cos x +C \end{equation}$

$\begin{equation} 84.\,\int\!\! \cos x dx= \sin x +C \end{equation}$

$\begin{equation} 85.\,\int\!\! \tan x dx -\ln \vert \cos x \vert +C \end{equation}$

$\begin{equation} 86.\,\int\!\! ctgx dx =\ln \vert \sin x \vert +C \end{equation}$

$\begin{equation} 87.\,\int\!\! \sec x dx = \ln \vert \tan (\frac{ \pi}{4} + \frac{x}{2}) \vert + C = \ln \vert \sec x + \tan x \vert +C \end{equation}$

$\begin{equation} 88.\,\int\!\! \csc x dx= \ln \vert \tan \frac{x}{2} \vert +C = \ln \vert \csc x - ctg x \vert +C \end{equation}$

$\begin{equation} 89.\,\int\!\! \sec^2 x dx = \tan x +C \end{equation}$

$\begin{equation} 90.\,\int\!\! \csc ^2 x dx = - ctgx +C \end{equation}$

$\begin{equation} 91.\,\int\!\! \sec x \tan x dx = \sec x +C \end{equation}$

$\begin{equation} 92.\,\int\!\! \csc x dx ctgx dx = -\csc x +C \end{equation}$

$\begin{equation} 93.\,\int\!\! \sin ^2 x dx = \frac{x}{2}- \frac{1}{4}\sin 2x +C \end{equation}$

$\begin{equation} 94.\,\int\!\! \cos ^2 x dx = \frac{x}{2} + \frac{1}{4}\sin 2x +C \end{equation}$

$\begin{equation} 95.\,\int\!\! \sin ^n x dx = - \frac{1}{n}\sin ^{n-1}x \cos x + \frac{n-1}{n} \int\!\! \sin ^{n-2}dx \end{equation}$

$\begin{equation} 96.\,\int\!\! \cos ^ n x dx = \frac{1}{n}\cos^{ n-1}x \sin x + \frac{n-1}{n} \int\!\! \cos^{n-2} x dx \end{equation}$

$\begin{equation} 97.\,\int\!\! \frac{dx}{\sin ^ n x} = - \frac{1}{n-1} . \frac{\cos x}{\sin ^{n-1}x}+\frac{n-2}{n-1} \int\!\! \frac{dx}{\sin^{n-2}x} \end{equation}$

$\begin{equation} 98.\,\int\!\! \frac{dx}{\cos ^n x}= \frac{1}{n-1}.\frac{\sin x}{\cos ^{n-1}x}+\frac{n-2}{n-1}\int\!\! \frac{dx}{\cos x^{n-2}x} \end{equation}$

$\begin{equation} 99.\,\int\!\! \cos ^ m \sin ^n x dx =\frac{1}{m+n}\cos^{m-1}x \sin ^{n+1}x + \frac{m-1}{m+n} \int\!\! \cos ^ {m-2} x \sin ^n x dx \end{equation}$

$\begin{equation} \qquad = -\frac{1}{m+1}\cos ^{m+1}x \sin ^{n-1}x + \frac{n-1}{m+n} \int\!\! \cos ^m x \sin ^{n-2} x dx \notag \end{equation}$

$\begin{equation} 100.\,\int\!\! \sin ax \cos bx dx = - \frac{1}{2(a+b)}\cos (a+b)x - \frac{1}{2(a-b)} \cos (a-b)x +C \end{equation}$

$\begin{equation} 101.\,\int\!\! \sin ax \sin bx dx = - \frac{1}{2(a+b)} \sin (a+b) x\frac{1}{2(a-b)} \sin (a-b)x +C \end{equation}$

$\begin{equation} 102.\,\int\!\! \cos ax \cos bx dx =\frac{1}{2(a+b)} \sin (a+b)x + \frac{1}{2(a-b)} \sin (a-b)x +C \end{equation}$

$\begin{equation} 103.\,\int\!\! \frac{dx}{a+b\sin x} = \frac{2}{\sqrt{a^2-b^2}}\arctan \frac{\arctan \frac{x}{2}+b}{\sqrt{a^2-b^2}} +C \qquad ( a^2 > b^2 ) \end{equation}$

$\begin{equation} 104.\,\int\!\! \frac{dx}{a+b \sin x} = \frac{1}{\sqrt{b^2-a^2}} \ln \Bigg \vert \frac{\arctan \frac{x}{2}+b - \sqrt{b^2-a^2}}{ \arctan \frac {x}{2}+b+ \sqrt {b^2-a^2}} \Bigg \vert +C \qquad (a^2<b^2) \end{equation}$

$\begin{equation} 105.\,\int\!\! \frac{dx}{a+b \cos x} = \frac{2}{a+b} \sqrt{ \frac{a+b}{a-b}} \arctan \Bigg ( \sqrt { \frac{a-b}{a+b}} \tan \frac{x}{2} \Bigg ) +C \qquad (a^2>b^2) \end{equation}$

$\begin{equation} 106.\,\int\!\! \frac{dx}{a+b \cos x}= \frac{1}{a+b}\sqrt{ \frac{a+b}{b-a}} \ln \Bigg \vert \frac{\tan \frac{x}{2}+ \sqrt {\frac{a+b}{b-a}}}{\tan \frac{x}{2}- \sqrt{\frac{a+b}{b-a}}} \Bigg \vert +C \qquad (a^2<b^2) \end{equation}$

$\begin{equation} 107.\,\int\!\! \frac{dx}{a^2\cos ^2x + b^2 \sin ^2 x}= \frac{1}{ab} \arctan (\frac{b}{a}\tan x ) +C \end{equation}$

$\begin{equation} 108.\,\int\!\! \frac{dx}{a^2 \cos ^2x -b^2 \sin ^2 x} = \frac{1}{2ab} \ln \Big \vert \frac{b \tan x +a }{b \tan x -a} \Big \vert +C \end{equation}$

$\begin{equation} 109.\,\int\!\! x \sin ax dx = \frac{1}{a^2} \sin ax - \frac{1}{a} x \cos ax +C \end{equation}$

$\begin{equation} 110.\,\int\!\! x^2 \sin ax dx = -\frac{1}{a}x^2 \cos ax + \frac{2}{a^2}x\sin ax + \frac{2}{a^3}\cos ax +C \end{equation}$

$\begin{equation} 111.\,\int\!\! x \cos ax dx = \frac{1}{a^2} \cos ax + \frac{1}{a}x \sin ax +C \end{equation}$

$\begin{equation} 112.\,\int\!\! x^2 \cos ax dx = \frac{1}{a} x^2 \sin ax + \frac{2}{a^2}x \cos ax - \frac{2}{a^3}\sin ax +C \end{equation}$

含有反三角函数的积分 (其中 $a > 0$ )

$\begin{equation} 113.\,\int\!\! \arcsin \frac{x}{a} dx = x \arcsin \frac{x}{a}+ \sqrt{a^2-x^2} +C \end{equation}$

$\begin{equation} 114.\,\int\!\! x\arcsin \frac{x}{a} dx = \Big ( \frac{x^2}{2}-\frac{a^2}{4}\Big )\arcsin \frac{x}{a} + \frac{x}{4}\sqrt {a^2-x^2} +C \end{equation}$

$\begin{equation} 115.\,\int\!\! x^2 \arcsin \frac{x}{a}dx= \frac{x^3}{3} \arcsin \frac{x}{a} + \frac{1}{9}(x^2+2a^2)\sqrt{a^2-x^2}+C \end{equation}$

$\begin{equation} 116.\,\int\!\! \arccos \frac{x}{a}dx= x \arccos \frac{x}{a}-\sqrt {a^2-x^2} +C \end{equation}$

$\begin{equation} 117.\,\int\!\! x \arccos \frac{x}{a} dx = \Big ( \frac{x^2}{2}-\frac{a^2}{4}\Big) \arccos \frac{x}{4} - \frac{x}{4}\sqrt {a^2 -x^2} +C \end{equation}$

$\begin{equation} 118.\,\int\!\! x^2\arccos \frac{x}{a} dx = \frac{x^3}{a} \arccos \frac{x}{a} - \frac{1}{9}(x^2 +2a^2)\sqrt{a^2-x^2}+C \end{equation}$

$\begin{equation} 119.\,\int\!\! \arctan \frac{x}{a} dx =x \arctan \frac{x}{a} - \frac{a}{2}\ln(a^2+x^2)+C \end{equation}$

$\begin{equation} 120.\,\int\!\! x\arctan \frac{x}{a}dx = \frac{1}{2}(a^2+x^2)\arctan \frac{x}{a}-\frac{a}{2}x+C \end{equation}$

$\begin{equation} 121.\,\int\!\! x^2 \arctan \frac{x}{a}dx = \frac{x^3}{3}\arctan \frac{x}{a} - \frac{a}{6}x^2+ \frac{a^3}{6}\ln (a^2+x^2)+C \end{equation}$

含有指数函数的积分

$\begin{equation} 122.\,\int\!\! a^x dx = \frac{1}{\ln a} a^x +C \end{equation}$

$\begin{equation} 123.\,\int\!\! e^{ax}dx = \frac{1}{a} e ^{ax}+C \end{equation}$

$\begin{equation} 124.\,\int\!\! xe^{ax}dx=\frac{1}{a^2}(ax-1)e^{ax}+c \end{equation}$

$\begin{equation} 125.\,\int\!\! x^n e^{ax}dx=\frac{1}{a}x^n e^{ax}- \frac{n}{a}\int\!\! x^{n-1} e^{ax}dx \end{equation}$

$\begin{equation} 126.\,\int\!\! xa^x dx= \frac{x}{\ln a}a^x - \frac{1}{(\ln a)^2}a^x+C \end{equation}$

$\begin{equation} 127.\,\int\!\! x^n a^x dx=\frac{1}{\ln a}x^na^x - \frac{n}{\ln a}\int\!\! x^{n-1} a^x dx \end{equation}$

$\begin{equation} 128.\,\int\!\! e^{ax} \sin bx dx = \frac{1}{a^2+b^2}e^{ax}(a\sin bx -b \cos bx)+C \end{equation}$

$\begin{equation} 129.\,\int\!\! e^{ax}\cos bx dx = \frac{1}{a^2+b^2} e^{ax} (b \sin bx + a \cos bx) +C \end{equation}$

$\begin{equation} 130.\,\int\!\! e^{ax}\sin ^n bx dx = \frac{1}{a^2+b^2n^2}e^{ax}\sin ^{n-1}bx (a \sin bx - nb \cos bx) \end{equation}$

$\begin{equation} \qquad \qquad \qquad \qquad \qquad+ \frac{n(n-1)b^2}{a^2+b^2n^2}\int\!\! e^{ax}\sin ^{n-2}bx dx \notag \end{equation}$

$\begin{equation} 131.\,\int\!\! e^{ax} \cos ^n bx dx = \frac{1}{a^2+b^2n^2} e^{ax} \cos ^{n-1} bx (a\cos bx + nb \sin bx) \end{equation}$

$\begin{equation} \qquad \qquad \qquad \qquad \qquad + \frac{n(n-1)b^2}{a^2+b^2n^2} \int\!\! e^{ax} \cos ^{n-2}bx dx \notag \end{equation}$

含有对数函数的积分

$\begin{equation} 132. \, \int\!\!\ln x dx=x\ln x -x +C \end{equation}$

$\begin{equation} 133.\,\int\!\! \frac {dx}{x\ln x}= \ln \vert \ln x \vert +C \end{equation}$

$\begin{equation} 134. \,\int\!\! x^n\ln x dx=\frac{1}{n+1}x^{n+1}(\ln x - \frac{1}{n+1})+C \end{equation}$

$\begin{equation} 135. \,\int\!\! (\ln x)^n dx=x(\ln x)^n - n \int\!\!(\ln x)^{n-1} dx \end{equation}$

$\begin{equation} 136. \,\int\!\! x^m (\ln x)^n dx=\frac{1}{m+1}x^{m+1}(\ln x)^n - \frac {n}{m+1} \int\!\! x^m (\ln x) ^{n-1} dx \end{equation}$

含有双曲函数的积分

$\begin{equation} 137. \,\int\!\! \sinh x dx = \cosh x +C \end{equation}$

$\begin{equation} 138. \,\int\!\! \cosh x dx = \sinh x +C \end{equation}$

$\begin{equation} 139. \,\int\!\! th x dx =\ln \cosh x +C \end{equation}$

$\begin{equation} 140. \,\int\!\! \sinh ^2 x dx = -\frac{x}{2} + \frac{1}{4} \sinh 2x +C \end{equation}$

$\begin{equation} 141. \,\int\!\! \cosh^2 x dx = \frac{x}{2}+\frac{1}{4} \sinh 2x +C \end{equation}$

定积分

$\begin{equation} 142. \int^{\pi}_{-\pi} \cos nx dx = \int^{\pi}_{\pi}\sin nx dx=0 \end{equation}$

$\begin{equation} 143. \,\int^ {\pi} _{-\pi} \cos mx \sin nx dx =0 \end{equation}$

$\begin{equation} 144. \,\int^{\pi} _{-\pi} \cos mx \cos nx dx =\left \{ \begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right. \end{equation}$

$\begin{equation} 145. \,\int ^\pi _{-\pi} \sin mx \sin nx dx = \left \{ \begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right. \end{equation}$

$\begin{equation} 146. \,\int^\pi _0 \sin mx \sin nx dx= \int ^\pi _0 \cos mx \cos nx dx = \left \{ \begin{array}{cc} 0, & m \neq n \\ \pi/2, & m=n \end{array} \right. \end{equation}$

$\begin{equation} 147. I_n =\,\int ^{\frac{\pi}{2}} _0 \sin ^n x dx = \int ^{\frac{\pi}{2}} _0 \cos ^n x dx \end{equation}$

$\begin{equation} \qquad I_n= \frac{n-1}{n} I _{n-2} \notag \end{equation}$

$\begin{equation} \qquad \left \{ \begin{aligned} I_n&= \frac{n-1}{n}.\frac{n-3}{n-2}.\dots \frac{4}{5} .\frac{2}{3} \quad \text{(n为大于1的正奇数)},I_1=1 \\ I_n&= \frac{n-1}{n} . \frac{n-3}{n-2} . \dots \frac{3}{4} .\frac{1}{2}. \frac{\pi}{2} \quad \text{(n为正偶数)},I_0= \frac{\pi}{2} \end{aligned} \right.\notag \end{equation}$