原始関数の一覧

テンプレート:出典の明記 本項は、原始関数の一覧（げんしかんすうのいちらん）である。以下、積分定数は$C$とする。

$ax+b$ を含む積分

$\int \frac{1}{ax+b}dx=\frac{1}{a}\ln |ax+b|+C$

$\int \frac{x}{ax+b}dx=\frac{x}{a}-\frac{b}{a^2}\ln |ax+b|+C$

$\int \frac{x^2}{ax+b}dx=\frac{1}{2a^3}(a^2{x}^2-2abx+2b^2\ln |ax+b|)+C$

$\int \frac{1}{x(ax+b)}dx=-\frac{1}{b}\ln \left|\frac{ax+b}{x}\right|+C$

$\int \frac{1}{x^2(ax+b)}dx=\frac{a}{b^2}\ln \left|\frac{ax+b}{x}\right|-\frac{1}{bx}+C$

$\sqrt{a+bx}$ を含む積分

$\int x\sqrt{a+bx}dx=\frac{2}{15b^2}(3bx-2a)(a+bx{)}^{\frac{3}{2}}+C$

$\int x^2\sqrt{a+bx}dx=\frac{2}{105b^3}(15b^2{x}^2-12abx+8a^2)(a+bx{)}^{\frac{3}{2}}+C$

$\int x^n\sqrt{a+bx}dx=\frac{2}{b(2n+3)}{x}^n(a+bx{)}^{\frac{3}{2}}-\frac{2na}{b(2n+3)}\int x^{n-1}\sqrt{a+bx}dx$

$\int \frac{\sqrt{a+bx}}{x}dx=2\sqrt{a+bx}+a\int \frac{1}{x\sqrt{a+bx}}dx$

$\int \frac{\sqrt{a+bx}}{x^n}dx=\frac{-1}{a(n-1)}\frac{(a+bx{)}^{\frac{3}{2}}}{x^{n-1}}-\frac{(2n-5)b}{2a(n-1)}\int \frac{\sqrt{a+bx}}{x^{n-1}}dx,n\ne 1$

$\int \frac{1}{x\sqrt{a+bx}}dx=\frac{1}{\sqrt{a}}\ln \left(\frac{\sqrt{a+bx}-\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right)+C,a>0$

$=\frac{2}{\sqrt{-a}}\arctan \sqrt{\frac{a+bx}{-a}}+C,a<0$

$\int \frac{1}{x^n\sqrt{a+bx}}dx=\frac{-1}{a(n-1)}\frac{\sqrt{a+bx}}{x^{n-1}}-\frac{(2n-3)b}{2a(n-1)}\int \frac{1}{x^{n-1}}\sqrt{a+bx}dx,n\ne 1$

$x^2\pm {\alpha }^2(\alpha \ne 0)$ を含む積分

$\int \frac{1}{x^2+{\alpha }^2}dx=\frac{1}{\alpha }\arctan \frac{x}{\alpha }+C$

$\int \frac{1}{\pm x^2\mp {\alpha }^2}dx=\frac{1}{2\alpha }\ln \left(\frac{x\mp \alpha }{\pm x+\alpha }\right)+C$

$a{x}^2+b$ を含む積分

$\int \frac{1}{a{x}^2+b}dx=\frac{1}{\sqrt{ab}}\arctan \sqrt{\frac{a}{b}}x+C$

$a{x}^2+bx+c(a\ne 0)$を含む積分

$\int (a{x}^2+bx+c)dx=\frac{a{x}^3}{3}+\frac{b{x}^2}{2}+cx+C$

$\sqrt{a^2+x^2}(a>0)$ を含む積分

$\int \sqrt{a^2+x^2}dx=\frac{1}{2}x\sqrt{a^2+x^2}+\frac{1}{2}{a}^2\ln (x+\sqrt{a^2+x^2})+C$

$\int x^2\sqrt{a^2+x^2}dx=\frac{1}{8}x(a^2+2x^2)\sqrt{a^2+x^2}-\frac{1}{8}{a}^4\ln (x+\sqrt{a^2+x^2})+C$

$\int \frac{\sqrt{a^2+x^2}}{x}dx=\sqrt{a^2+x^2}-a\ln \left(\frac{a+\sqrt{a^2+x^2}}{x}\right)+C$

$\int \frac{\sqrt{a^2+x^2}}{x^2}dx=\ln (x+\sqrt{a^2+x^2})-\frac{\sqrt{a^2+x^2}}{x}+C$

$\int \frac{1}{\sqrt{a^2+x^2}}dx=\ln (x+\sqrt{a^2+x^2})+C$

$\int \frac{x^2}{\sqrt{a^2+x^2}}dx=\frac{1}{2}x\sqrt{a^2+x^2}-\frac{1}{2}{a}^2\ln (\sqrt{a^2+x^2}+x)+C$

$\int \frac{1}{x\sqrt{a^2+x^2}}dx=\frac{1}{a}\ln \left(\frac{x}{a+\sqrt{a^2+x^2}}\right)+C$

$\int \frac{1}{x^2\sqrt{a^2+x^2}}dx=-\frac{\sqrt{a^2+x^2}}{a^{2}x}+C$

$\sqrt{x^2-a^2}(x^2>a^2)$ を含む積分

$\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln (x+\sqrt{x^2-a^2})+C$

$\sqrt{a^2-x^2}(a^2>x^2)$ を含む積分

$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \frac{x}{a}+C=-\arccos \frac{x}{a}+C$

$\int \sqrt{a^2-x^2}dx=\frac{1}{2}x\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin \frac{x}{a}+C$

$\int x^2\sqrt{a^2-x^2}dx=\frac{1}{8}x(2x^2-a^2)\sqrt{a^2-x^2}+\frac{1}{8}{a}^4\arcsin \frac{x}{a}+C$

$\int \frac{\sqrt{a^2-x^2}}{x}dx=\sqrt{a^2-x^2}-a\ln \left(\frac{a+\sqrt{a^2-x^2}}{x}\right)+C$

$\int \frac{\sqrt{a^2-x^2}}{x^2}dx=-\frac{\sqrt{a^2-x^2}}{x}-\arcsin \frac{x}{a}+C$

$\int \frac{1}{x\sqrt{a^2-x^2}}dx=-\frac{1}{a}\ln \left(\frac{a+\sqrt{a^2-x^2}}{x}\right)+C$

$\int \frac{x^2}{\sqrt{a^2-x^2}}dx=-\frac{1}{2}x\sqrt{a^2-x^2}+\frac{1}{2}{a}^2\arcsin \frac{x}{a}+C$

$\int \frac{1}{x^2\sqrt{a^2-x^2}}dx=-\frac{\sqrt{a^2-x^2}}{a^{2}x}+C$

$R=\sqrt{|a|x^2+bx+c}(a\ne 0)$ を含む積分

$\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\ln (2\sqrt{a}R+2ax+b)(\text{for }a>0)$

$\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\mathrm{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(for }a>0\text{, }4ac-b^2>0\text{)}$

$\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\ln |2ax+b|\text{(for }a>0\text{, }4ac-b^2=0\text{)}$

$\int \frac{dx}{R}=-\frac{1}{\sqrt{-a}}\arcsin \frac{2ax+b}{\sqrt{b^2-4ac}}\text{(for }a<0\text{, }4ac-b^2<0\text{, }(2ax+b)<\sqrt{b^2-4ac}\text{)}$

$\int \frac{dx}{R^3}=\frac{4ax+2b}{(4ac-b^2)R}$

$\int \frac{dx}{R^5}=\frac{4ax+2b}{3(4ac-b^2)R}(\frac{1}{R^2}+\frac{8a}{4ac-b^2})$

$\int \frac{dx}{R^{2n+1}}=\frac{2}{(2n-1)(4ac-b^2)}(\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int \frac{dx}{R^{2n-1}})$

$\int \frac{x}{R}dx=\frac{R}{a}-\frac{b}{2a}\int \frac{dx}{R}$

$\int \frac{x}{R^3}dx=-\frac{2bx+4c}{(4ac-b^2)R}$

$\int \frac{x}{R^{2n+1}}dx=-\frac{1}{(2n-1)a{R}^{2n-1}}-\frac{b}{2a}\int \frac{dx}{R^{2n+1}}$

$\int \frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln \left(\frac{2\sqrt{c}R+bx+2c}{x}\right)$

$\int \frac{dx}{xR}=-\frac{1}{\sqrt{c}}\mathrm{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)$

三角関数を含む積分

$\int \cos xdx=\sin x+C$

$\int -\sin xdx=\cos x+C$

$\int {\sec }^{2}xdx=\tan x+C$

$\int -{\csc }^{2}xdx=\cot x+C$

$\int \sec x\tan xdx=\sec x+C$

$\int -\csc x\cot xdx=\csc x+C$

$\int \tan xdx=-\ln (\cos x)+C$

$\int \cot xdx=\ln (\sin x)+C$

$\int \sec xdx=\ln (\sec x+\tan x)+C={\mathrm{gd}}^{-1}x+C{\mathrm{gd}}^{-1}x$：グーデルマン関数の逆関数

$\int \csc xdx=-\ln (\csc x+\cot x)+C=\ln \left(\frac{\tan x-\sin x}{\sin x\tan x}\right)+C$

$\int {\sin }^{n}xdx=-\frac{1}{n}{\sin }^{n-1}x\cos x+\frac{n-1}{n}\int {\sin }^{n-2}xdx+C\mathrm{\forall }n\ge 2$

$\int {\sin }^{2}xdx=\frac{x}{2}-\frac{\sin 2x}{4}+C$

$\int {\cos }^{n}xdx=\frac{1}{n}{\cos }^{n-1}x\sin x+\frac{n-1}{n}\int {\cos }^{n-2}xdx+C\mathrm{\forall }n\ge 2$

$\int {\cos }^{2}xdx=\frac{x}{2}+\frac{\sin 2x}{4}+C$

$\int {\tan }^{n}xdx=\frac{1}{n-1}{\tan }^{n-1}x-\int {\tan }^{n-2}xdx+C\mathrm{\forall }n\ge 2$

$\int {\tan }^{2}xdx=\tan x-x+C$

$\int {\cot }^{n}xdx=\frac{1}{n-1}{\cot }^{n-1}x-\int {\cot }^{n-2}xdx+C\mathrm{\forall }n\ge 2$

$\int {\cot }^{2}xdx=-\cot x-x+C$

$\int {\sec }^{n}xdx=\frac{1}{n-1}{\sec }^{n-2}x\tan x+\frac{n-2}{n-1}\int {\sec }^{n-2}xdx+C\mathrm{\forall }n\ge 2$

$\int {\csc }^{n}xdx=-\frac{1}{n-1}{\csc }^{n-2}x\cot x+\frac{n-2}{n-1}\int {\csc }^{n-2}xdx+C\mathrm{\forall }n\ge 2$

逆三角関数を含む積分

$\int \arcsin xdx=x\arcsin x+\sqrt{1-x^2}+C$

$\int \arccos xdx=x\arccos x-\sqrt{1-x^2}+C$

$\int \arctan xdx=x\arctan x-\ln \sqrt{1+x^2}+C$

$\int \mathrm{arccot}xdx=x\mathrm{arccot}x+\ln \sqrt{1+x^2}+C$

$\int \mathrm{arcsec}xdx=x\mathrm{arcsec}x-\ln (x-\sqrt{x^2-1})+C$

$\int \mathrm{arccsc}xdx=x\mathrm{arccsc}x+\ln (x+\sqrt{x^2-1})+C$

指数関数を含む積分

$\int e^{x}dx=e^x+C$

$\int {\alpha }^{x}dx=\frac{{\alpha }^x}{\ln \alpha }+C$

$\int x{e}^{ax}dx=\frac{1}{a^2}(ax-1)e^{ax}+C$

$\int x^n{e}^{ax}dx=\frac{1}{a}{x}^n{e}^{ax}-\frac{n}{a}\int x^{n-1}{e}^{ax}dx$

$\int e^{ax}\sin bxdx=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C$

$\int e^{ax}\cos bxdx=\frac{e^{ax}}{a^2+b^2}(a\cos bx+b\sin bx)+C$

対数関数を含む積分

$\int \ln xdx=x\ln x-x+C$

$\int {\log }_{\alpha }xdx=\frac{1}{\ln \alpha }\left(x\ln x-x\right)+C$

$\int x^n\ln xdx=\frac{x^{n+1}}{(n+1{)}^2}[(n+1)\ln x-1]+C$

$\int \frac{1}{x\ln x}dx=\ln (\ln x)+C$

双曲線関数を含む積分

$\int \sinh xdx=\cosh x+C$

$\int \cosh xdx=\sinh x+C$

$\int \tanh xdx=\ln (\cosh x)+C$

$\int \coth xdx=\ln (\sinh x)+C$

$\int \text{sech}xdx=\arcsin (\tanh x)+C=\arctan (\sinh x)+C=\mathrm{gd}x+C\mathrm{gd}x$：グーデルマン関数

$\int \text{csch}xdx=\ln (\tanh \frac{x}{2})+C$

定積分

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-\alpha x^2}dx=\sqrt{\frac{\pi }{\alpha }}$

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\text{sin}}^{n}xdx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\text{cos}}^{n}xdx=\{\begin{array}{cc}\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \cdots \cdot \frac{4}{5}\cdot \frac{2}{3},& \text{if }n>1\text{ and }n\text{ is odd}\\ \frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \cdots \cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{\pi }{2},& \text{if }n>0\text{ and }n\text{ is even}\end{array}$

関連項目

• 微分積分学の基礎定理
• 解析学