积分表一（高等数学同济版中所有的积分公式）

文章目录

含有 $a x + b$ 的积分

$\begin{equation} 1.\,\int\!\! \frac{dx}{ax+b}=\frac{1}{a}\ln \vert ax+b \vert +C \end{equation}$

$\begin{equation} 2.\,\int\!\! (ax+b)^\mu dx=\frac{1}{a(\mu+1)}(ax+b)^{\mu+1}+C \qquad (\mu \neq -1) \end{equation}$

$\begin{equation} 3.\,\int\!\! \frac{x}{ax+b}dx=\frac{1}{a^2}(ax+b-b\ln\vert (x+b) \vert ) +C \end{equation}$

$\begin{equation} 4.\,\int\!\! \frac{x^2}{ax+b}dx=\frac{1}{a^3}\Big[\frac{1}{2}(ax+b)^2-2b(ax+b)+b^2\ln\vert ax+b\vert\Big]+C \end{equation}$

$\begin{equation} 5.\,\int\!\! \frac{dx}{x(ax+b)}=-\frac{1}{b}\ln \Big\vert \frac{ax+b}{x}\Big\vert +C \end{equation}$

$\begin{equation} 6.\,\int\!\! \frac{dx}{x^2(ax+b)}=-\frac{1}{bx}+\frac{a}{b^2}\ln\Big\vert\frac{ax+b}{x}\Big\vert+C \end{equation}$

$\begin{equation} 7.\,\int\!\! \frac{x}{(ax+b)^2}dx=\frac{1}{a^2}\Big(\ln \vert ax+b \vert + \frac{b}{ax+b}\Big)+C \end{equation}$

$\begin{equation} 8.\,\int\!\! \frac{x^2}{(ax+b)^2}dx= \frac{1}{a^3}\Big(ax+b-2b\ln\vert ax+b \vert - \frac{b^2}{ax+b}\Big ) + C \end{equation}$

$\begin{equation} 9.\,\int\!\! \frac{dx}{x(a+b)^2}=\frac {1}{b(ax+b)}-\frac{1}{b^2}\ln\Big\vert \frac{ax+b}{x}\Big\vert + C \end{equation}$

含有 $\sqrt{ax+b}$ 的积分

$\begin{equation} 10.\,\int\!\! \sqrt{ax+b} dx= \frac{2}{3a}\sqrt{(ax+b)^3}+C \end{equation}$

$\begin{equation} 11.\,\int\!\! x \sqrt{ax+b}dx=\frac{2}{15a^2}(3ax-2b)\sqrt{(ax+b)^3}+C \end{equation}$

$\begin{equation} 12.\,\int\!\! x^2 \sqrt{ax+b}dx=\frac{2}{105a^3}(15a^2x^2-12abx+8b^2)\sqrt{(ax+b)^3}+C \end{equation}$

$\begin{equation} 13.\,\int\!\! \frac{x}{\sqrt {ax+b}}dx= \frac{2}{3a^2}(ax-2b)\sqrt{ax+b}+C \end{equation}$

$\begin{equation} 14.\,\int\!\! \frac{x^2}{\sqrt {ax+b}}dx= \frac{2}{15a^2}(2a^2x^2-4abx+8b^2)\sqrt{ax+b}+C \end{equation}$

$\begin{equation} 15.\,\int\!\! \frac{dx}{x\sqrt{ax+b}}=\left \{ \begin{array}{cc} \!\!\!\!\frac{1}{\sqrt{b}}\ln \Big \vert \frac{ \sqrt{ax+b} - \sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \Big \vert + C & \textrm ( b>0) \\ \frac{2}{-\sqrt{-b}}\arctan \sqrt{\frac{ax+b}{-b}}+C & \textrm (b<0) \end{array} \right. \end{equation}$

$\begin{equation} 16.\,\int\!\! \frac{dx}{x^2\sqrt{ax+b}} = - \frac{\sqrt{ax+b}}{bx} - \frac{a}{2b}\int\!\! \frac{dx}{x\sqrt{ax+b}} \end{equation}$

$\begin{equation} 17.\,\int\!\! \frac{\sqrt{ax+b}}{x}dx = 2 \sqrt{ax+b}+b \int\!\! \frac{dx}{x\sqrt{ax+b}} \end{equation}$

$\begin{equation} 18.\,\int\!\! \frac{\sqrt{ax+b}}{x^2}dx= - \frac{\sqrt{ax+b}}{x}+ \frac{a}{2}\int\!\! \frac{dx}{x\sqrt{ax+b}} \end{equation}$

含有 $x^2 \pm a$ 的积分

$\begin{equation} 19.\,\int\!\! \frac{dx}{x^2+a^2}= \frac{1}{a} \arctan \frac{x}{a} +C \end{equation}$

$\begin{equation} 20.\,\int\!\! \frac{dx}{(x^2+a^2)^n} = \frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac {2n-3}{2(n-1)a^2} \int\!\! \frac {dx}{(x^2+ a^2)^{n-1}} \end{equation}$

$\begin{equation} 21.\,\int\!\!\ \frac{dx}{x^2-a^2}= \frac{1}{2a} \ln \Big \vert \frac{x-a}{x+a}\Big \vert + C \end{equation}$

含有 $ax^2 +b$ 的积分

$\begin{equation} 22.\,\int\!\! \frac{dx}{ax^2+b} = \left \{ \begin{array}{cc} \!\!\!\!\frac{1}{\sqrt{ab}}\arctan \sqrt {\frac {a}{b}}x +C & \textrm (b>0) \\ \frac{1}{2 \sqrt{-ab}} \ln\Big \vert \frac { \sqrt {a}x - \sqrt {-b}}{\sqrt{a}x+ \sqrt{-b}} \Big \vert + C & \textrm (b<0) \end{array} \right. \end{equation}$

$\begin{equation} 23.\,\int\!\! \frac{x} {ax^2 +b} dx = \frac{1}{2a} \ln \vert ax^2+b \vert +C \end{equation}$

$\begin{equation} 24.\,\int\!\! \frac{x^2}{ax^2+b} dx= \frac{x}{a} - \frac{b}{a} \int\!\! \frac{dx}{ax^2+b} \end{equation}$

$\begin{equation} 25.\,\int\!\! \frac {dx} { x(ax^2+b)} = \frac {1}{2b} \ln \frac {x^2}{ \vert ax^2+b \vert } +C \end{equation}$

$\begin{equation} 26.\,\int\!\! \frac {dx}{x^2(ax^2+b)} = - \frac{1}{bx} - \frac{a}{b} \int\!\! \frac {dx}{ax^2+b} \end{equation}$

$\begin{equation} 27.\,\int\!\! \frac{dx}{x^3(ax^2+b)} = \frac {a}{2b^2} \ln \frac { \vert ax^2+b\vert }{x^2} - \frac{1}{2bx^2} + C \end{equation}$

$\begin{equation} 28.\,\int\!\! \frac{dx}{(ax^2+b)^2} = \frac{x}{2b(ax^2+b)} + \frac{1}{2b} \int\!\! \frac {dx} { ax^2+b} \end{equation}$

含有 $ax^2+bx+c$ 的积分

$\begin{equation} 29.\,\int\!\! \frac{dx}{ax^2+bx+c}dx= \left \{ \begin{array}{cc} \!\!\! \frac{2}{\sqrt{4ac-b^2}} \arctan \frac {2ax+b}{\sqrt{4ac-b^2}}+C & \textrm (b^2<4ac) \\ \frac {1}{\sqrt {b^2-4ac}} \ln \Big \vert \frac {2ax+b-\sqrt{b^2-4ac}} {2ax+b+ \sqrt{b^2-4ac}} \Big \vert +C & \textrm (b^2>4ac) \end{array} \right. \end{equation}$

$\begin{equation} 30.\,\int\!\! \frac{x}{ax^2+bx+c} dx = \frac{1}{2a}\ln \vert ax^2+bx+c \vert - \frac{b}{2a} \int\!\! \frac {dx}{ax^2+bx+c} \end{equation}$

含有 $\sqrt{x^2+a^2} \quad (a>0)$ 的积分

$\begin{equation} 31.\,\int\!\! \frac{dx}{\sqrt{x^2+a^2}} = arsh \frac{x}{a}+C1= \ln (x+\sqrt{x^2+a^2})+C \end{equation}$

$\begin{equation} 32.\,\int\!\! \frac{dx}{\sqrt{(x^2+a^2)^3}}= \frac{x}{a^2\sqrt{x^2+a^2}}+C \end{equation}$

$\begin{equation} 33.\,\int\!\! \frac{x}{\sqrt{x^2+a^2}}dx=\sqrt{x^2+a^2}+C \end{equation}$

$\begin{equation} 34.\,\int\!\! \frac{x}{\sqrt{(x^2+a^2)^3}}dx= - \frac{1}{\sqrt{x^2+a^2}}+C \end{equation}$

$\begin{equation} 35.\,\int\!\! \frac{x^2}{\sqrt{x^2+a^2}}dx = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C \end{equation}$

$\begin{equation} 36.\,\int\!\! \frac{x^2}{\sqrt{(x^2+a^2)^3}}dx= -\frac{x}{\sqrt{x^2+a^2}}+\ln (x+\sqrt{x^2+a^2})+C \end{equation}$

$\begin{equation} 37.\,\int\!\! \frac{dx}{x\sqrt{x^2+a^2}}= \frac{1}{a}\ln \frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C \end{equation}$

$\begin{equation} 38.\,\int\!\! \frac{dx}{x^2\sqrt{x^2+a^2}}= -\frac{\sqrt{x^2+a^2}}{a^2x}+C \end{equation}$

$\begin{equation} 39.\,\int\!\! \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C \end{equation}$

$\begin{equation} 40.\,\int\!\! \sqrt{(x^2+a^2)^3}dx = \frac{x}{8}(2x^2+5a^2)\sqrt{x^2+a^2}+ \frac{3}{8}a^4\ln(x+\sqrt{x^2+a^2})+C \end{equation}$

$\begin{equation} 41.\,\int\!\! x\sqrt{x^2+a^2}dx= \frac{1}{3}\sqrt{(x^2+a^2)^3}+C \end{equation}$

$\begin{equation} 42.\,\int\!\! x^2 \sqrt{x^2+a^2}dx=\frac{x}{8}(2x^2+a^2)\sqrt{x^2+a^2}-\frac{a^4}{8}\ln(x+\sqrt{x^2+a^2})+C \end{equation}$

$\begin{equation} 43.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x}dx= \sqrt{x^2+a^2}+a\ln\frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C \end{equation}$

$\begin{equation} 44.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x^2}dx= -\frac{\sqrt{x^2+a^2}}{x} + \ln(x+\sqrt{x^2+a^2}) +C \end{equation}$

含有 $\sqrt{x^2-a^2} \quad (a>0)$ 的积分

$\begin{equation} 45.\,\int\!\! \frac{dx}{\sqrt{x^2-a^2}}= \frac{x}{\vert x \vert} arch \frac{\vert x \vert}{a}+C_1= ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation}$

$\begin{equation} 46.\,\int\!\! \frac{dx}{\sqrt{(x^2-a^2)^3}}= - \frac{x}{a^2\sqrt{x^2-a^2}} +C \end{equation}$

$\begin{equation} 47.\,\int\!\! \frac{x}{\sqrt{x^2-a^2}}dx = \sqrt{x^2-a^2}+C \end{equation}$

$\begin{equation} 48.\,\int\!\! \frac{x}{\sqrt{(x^2-a^2)^3}}dx = - \frac{1}{\sqrt{x^2-a^2}} +C \end{equation}$

$\begin{equation} 49.\,\int\!\! \frac{x^2}{\sqrt{x^2-a^2}}dx =\frac{x}{2}\sqrt{x^2 -a ^2} + \frac{a^2}{2} \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation}$

$\begin{equation} 50.\,\int\!\! \frac{x^2}{\sqrt{(x^2-a^2)^3}}dx =-\frac{x}{\sqrt{x^2 -a ^2}} + \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation}$

$\begin{equation} 51.\,\int\!\! \frac{dx}{x\sqrt{x^2-a^2}}= \frac{1}{a} \arccos \frac{a}{\vert x \vert} +C \end{equation}$

$\begin{equation} 52.\,\int\!\! \frac{dx}{x^2\sqrt{x^2-a^2}} =\frac{ \sqrt {x^2-a^2}}{a^2x}+C \end{equation}$

$\begin{equation} 53.\,\int\!\! \sqrt{x^2-a^2}dx = \frac{x}{2} \sqrt{x^2-a^2} - \frac{a^2}{2} \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation}$

$\begin{equation} 54.\,\int\!\! \sqrt{(x^2-a^2)^3}dx =\frac{x}{8}(2x^2-5a^2)\sqrt{x^2-a^2} + \frac{3}{8} a^4 \ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation}$

$\begin{equation} 55.\,\int\!\! x \sqrt{x^2-a^2}dx= \frac{1}{3}\sqrt{(x^2-a^2)^3} +C \end{equation}$

$\begin{equation} 56.\,\int\!\! x^2 \sqrt{x^2-a^2} dx = \frac{x}{8} (2x^2-a^2)\sqrt{x^2-a^2}-\frac{a^4}{8}\ln \vert x+ \sqrt{x^2-a^2} \vert +C \end{equation}$

$\begin{equation} 57.\,\int\!\! \frac{\sqrt{x^2-a^2}}{x} dx= \sqrt{x^2-a^2}- \arccos \frac{a}{\vert x \vert} +C \end{equation}$

$\begin{equation} 58.\,\int\!\! \frac{\sqrt{x^2-a^2}}{x^2} dx = -\frac{\sqrt{x^2-a^2}}{x} + \ln \vert x + \sqrt{x^2-a^2} \vert +C \end{equation}$

含有 $\sqrt{a^2-x^2} \quad (a>0)$ 的积分

$\begin{equation} 59.\,\int\!\! \frac{dx}{\sqrt{a^2-x^2}} =\arcsin \frac{x}{a} +C \end{equation}$

$\begin{equation} 60.\,\int\!\! \frac{dx}{\sqrt{(a^2-x^2)^3}}= \frac{x}{a^2\sqrt{a^2-x^2}}+C \end{equation}$

$\begin{equation} 61.\,\int\!\! \frac{x}{\sqrt{a^2-x^2}} dx = - \sqrt{a^2-x^2}+C \end{equation}$

$\begin{equation} 62.\,\int\!\! \frac{x}{\sqrt{(a^2-x^2)^3}} dx = - \frac{1}{\sqrt{a^2-x^2}} +C \end{equation}$

$\begin{equation} 63.\,\int\!\! \frac{x^2}{\sqrt{a^2-x^2}}dx = -\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin \frac{x}{a}+C \end{equation}$

$\begin{equation} 64.\,\int\!\! \frac{x^2}{\sqrt{(a^2-x^2)^3}}dx =\frac{x}{\sqrt{a^2-x^2}}- \arcsin \frac{x}{a} +C \end{equation}$

$\begin{equation} 65.\,\int\!\! \frac{dx}{x\sqrt{a^2-x^2}}=\frac{1}{a}\ln\frac{a-\sqrt{a^2-x^2}}{\vert x \vert } +C \end{equation}$

$\begin{equation} 66.\,\int\!\! \frac{dx}{x^2\sqrt{a^2-x^2}}=- \frac{a^2-x^2}{a^2x}+C \end{equation}$

$\begin{equation} 67.\,\int\!\! \sqrt{a^2-x^2}dx= \frac{x}{2} \sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin \frac{x}{a}+C \end{equation}$

$\begin{equation} 68.\,\int\!\! \sqrt{(a^2-x^2)^3}dx=\frac{x}{8}(5a^2-2x^2)\sqrt{a^2-x^2}+\frac{3}{8}a^4\arcsin\frac{x}{a}+C \end{equation}$

$\begin{equation} 69.\,\int\!\! x\sqrt{a^2-x^2}dx=-\frac{1}{3}\sqrt{(a^2-x^2)^3}+C \end{equation}$

$\begin{equation} 70.\,\int\!\! x^2\sqrt{a^2-x^2}dx=\frac{x}{8}(2x^2-a^2)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin\frac{x}{a}+C \end{equation}$

$\begin{equation} 71.\,\int\!\! \frac{\sqrt{a^2-x^2}}{x} dx =\sqrt{a^2-x^2}+a \ln \frac{a-\sqrt{a^2-x^2}}{\vert x \vert}+C \end{equation}$

$\begin{equation} 72.\,\int\!\! \frac{\sqrt{a^2-x^2}}{x^2}dx=-\frac{\sqrt{a^2-x^2}}{x}-\arcsin \frac{x}{a}+C \end{equation}$

含有 $\sqrt{\pm ax^2+bx+c} \quad (a>0)$ 的积分

$\begin{equation} 73.\,\int\!\! \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}\ln \vert 2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\,\vert +C \end{equation}$

$\begin{equation} 74.\,\int\!\! \sqrt{ax^2+bx+c}\,dx=\frac{2ax+b}{4a}\sqrt{ax^2+bx+c} +\frac{4ac-b^2}{8\sqrt{a^3}}\ln\vert 2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\vert +C \end{equation}$

$\begin{equation} 75.\,\int\!\! \frac{x}{\sqrt{ax^2 + bx + c}}dx = \frac{1}{a}\sqrt{ ax^2 + bx + c} -\frac{b}{2\sqrt{a^3}}\ln \vert 2ax + b + 2 \sqrt{a}\sqrt{ax^2 + bx + c}\,\vert +C \end{equation}$

$\begin{equation} 76.\,\int\!\! \frac{dx}{\sqrt{c+bx-ax^2}}= \frac{1}{\sqrt{a}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C \end{equation}$

$\begin{equation} 77.\,\int\!\! \sqrt{c+bx-ax^2}\,dx=\frac{2ax-b}{4a}\sqrt{c+bx-ax^2}+\frac{b^2+4ac}{8\sqrt{a^3}}\arcsin \frac{2ax-b}{\sqrt{b^2+4ac}}+C \end{equation}$

$\begin{equation} 78.\,\int\!\! \frac{x}{\sqrt{c+bx-ax^2}}\,dx=-\frac{1}{a}\sqrt{c+bx-ax^2}+\frac{b}{2\sqrt{a^3}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C \end{equation}$