原始関数の一覧

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、積分定数はとする。

## を含む積分
:\int\frac{1}{ax+b}\,dx=\frac{1}{a}\ln \left | ax+b\right|+C
:\int\frac{x}{ax+b}\,dx=\frac{x}{a}-\frac{b}{a^2}\ln\left |ax+b \right|+C
:\int\frac{x^2}{ax+b}\,dx=\frac{1}{2a^3}(a^2x^2-2abx+2b^2 \ln\left |ax+b\right|)+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

## 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^2x^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\neq 1
:\begin{align}
\int\frac{1}{x\sqrt{a+bx}}\,dx=\begin{cases}
\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
:\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\neq 1

## 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(\dfrac{x\mp\alpha}{\pm x+\alpha}\right)+C

## を含む積分
:\int\frac{1}{ax^2+b}\,dx=\frac{1}{\sqrt{ab}} \arctan\sqrt{\frac{a}{b}}x+C

## を含む積分
:\int (ax^2+bx+c)\,dx=\frac{ax^3}{3}+\frac{bx^2}{2}+cx+C

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

## を含む積分
:\int \frac{1}{\sqrt{x^2-a^2}}\,dx=\ln\left ( x+\sqrt{x^2-a^2} \right ) +C

## 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^2x}+C

## を含む積分
: \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left(2\sqrt{a}R+2ax+b\right)\qquad(\mbox{for }a>0)

: \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\,\operatorname{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad \mbox{(for }a>0\mbox{, }4ac-b^2>0\mbox{)}

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

: \int\frac{dx}{R} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad \mbox{(for }a

: \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}\left(\frac{1}{R^2}+\frac{8a}{4ac-b^2}\right)

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

: \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)aR^{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}}\operatorname{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)

## 三角関数を含む積分

:\int\cos x\,dx=\sin x+C
:\int-\sin x\,dx=\cos x+C
:\int\sec^2x\,dx=\tan x+C
:\int-\csc^2x\,dx=\cot x+C
:\int\sec x\tan x\,dx=\sec x+C
:\int-\csc x\cot x\,dx=\csc x+C

:\int\tan x\,dx= - \ln(\cos x)+C
:\int\cot x\,dx=\ln(\sin x)+C
:\int\sec x\,dx=\ln(\sec x+\tan x)+C=\operatorname{gd}^{-1}x+C\quad\operatorname{gd}^{-1}x：グーデルマン関数の逆関数
:\int\csc x\,dx=-\ln(\csc x+\cot x)+C=\ln\left({\tan x-\sin x\over\sin x\tan x}\right)+C

:\int \sin ^n x \,dx = - \frac{1}{n} \sin ^{n-1} x \cos x + \frac{n-1}{n} \int \sin ^{n-2} x \,dx +C \quad \forall n \ge 2
:\int \sin ^2 x \,dx = \frac{x}{2}-\frac{\sin{2x}}{4} +C

:\int \cos ^n x \,dx = \frac{1}{n} \cos ^{n-1} x \sin x + \frac{n-1}{n} \int \cos ^{n-2} x \,dx +C \quad \forall n \ge 2
:\int \cos ^2 x \,dx = \frac{x}{2}+\frac{\sin{2x}}{4} +C

:\int \tan ^n x \,dx = \frac{1}{n-1} \tan ^{n-1} x - \int \tan ^{n-2} x \,dx +C \quad \forall n \ge 2
:\int \tan ^2 x \,dx = \tan x - x +C

:\int \cot ^n x \,dx = -\frac{1}{n-1} \cot ^{n-1} x - \int \cot ^{n-2} x \,dx +C \quad \forall n \ge 2
:\int \cot ^2 x \,dx = - \cot x - x +C

:\int \sec ^n x \,dx = \frac{1}{n-1} \sec ^{n-2} x \tan x + \frac{n-2}{n-1} \int \sec ^{n-2} x \,dx +C \quad \forall n \ge 2

:\int \csc ^n x \,dx = - \frac{1}{n-1} \csc ^{n-2} x \cot x + \frac{n-2}{n-1} \int \csc ^{n-2} x \,dx +C \quad \forall n \ge 2

## 逆三角関数を含む積分

:\int \arcsin x \,dx = x \arcsin x + \sqrt {1 - x^2} +C
:\int \arccos x \,dx = x \arccos x - \sqrt {1 - x^2} +C
:\int \arctan x \,dx = x \arctan x - \ln \sqrt {1 + x^2} +C
:\int \arccot x \,dx = x \arccot x + \ln \sqrt {1 + x^2} +C
:\int \arcsec x \,dx = x \arcsec x - \ln (x - \sqrt{x^2 - 1}) +C
:\int \arccsc x \,dx = x \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 xe^{ax}\,dx=\frac{1}{a^2}(ax-1)e^{ax}+C
:\int x^ne^{ax}\,dx=\frac{1}{a}x^ne^{ax}-\frac{n}{a}\int x^{n-1}e^{ax}\,dx
:\int e^{ax}\sin bx \,dx=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C
:\int e^{ax}\cos bx \,dx=\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 x\,dx = x\ln x - x + C
:\int\log_\alpha x\,dx=\frac{1}{\ln\alpha}\left({x\ln x - x}\right)+C
:\int x^n\ln x\,dx = \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 x \,dx = \cosh x +C
:\int \cosh x \,dx = \sinh x +C
:\int \tanh x \,dx = \ln\left(\cosh x\right) +C
:\int \coth x \,dx = \ln\left(\sinh x\right) +C
:\int \mbox{sech}\ x \,dx = \arcsin\left(\tanh x\right) + C =  \arctan\left(\sinh x\right) + C = \operatorname{gd}x+C\quad\operatorname{gd}x：グーデルマン関数
:\int \mbox{csch}\ x \,dx = \ln\left(\tanh {x \over2}\right) + C

## 定積分
:\int^\infty_{-\infty}e^{-\alpha x^2}\,dx=\sqrt{\frac{\pi}{\alpha}}
:\int_0^\frac{\pi}{2} \mbox{sin}^n x\,dx=\int_0^\frac{\pi}{2} \mbox{cos}^n x\,dx=
\begin{cases}
\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\cdots\cdot\frac{4}{5}\cdot\frac{2}{3}, & \mbox{if }n>1\mbox{ and }n\mbox{ 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}, & \mbox{if }n>0\mbox{ and }n\mbox{ is even}
\end{cases}

## 関連項目

* 微分積分学の基礎定理
* 解析学