무리함수 적분표

아래 목록은 무리 함수의 부정적분이다.

아래의 적분식들에서 적분상수는 생략하였다.

## ''r'' = 을 포함하는 함수의 적분

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

: \int r^3\,dx = \frac{1}{4}xr^3+\frac{3}{8}a^2xr+\frac{3}{8}a^4\ln\left(x+r\right)

: \int r^5\,dx = \frac{1}{6}xr^5+\frac{5}{24}a^2xr^3+\frac{5}{16}a^4xr+\frac{5}{16}a^6\ln\left(x+r\right)

: \int x r\,dx = \frac{r^3}{3}

: \int x r^3\,dx = \frac{r^5}{5}

: \int x r^{2n+1}\,dx = \frac{r^{2n+3}}{2n+3} 

: \int x^2 r\,dx = \frac{xr^3}{4}-\frac{a^2xr}{8}-\frac{a^4}{8}\ln\left(x+r\right)

: \int x^2 r^3\,dx = \frac{xr^5}{6}-\frac{a^2xr^3}{24}-\frac{a^4xr}{16}-\frac{a^6}{16}\ln\left(x+r\right)

: \int x^3 r\,dx = \frac{r^5}{5} - \frac{a^2 r^3}{3}

: \int x^3 r^3\,dx = \frac{r^7}{7}-\frac{a^2r^5}{5} 

: \int x^3 r^{2n+1}\,dx = \frac{r^{2n+5}}{2n+5} - \frac{a^2 r^{2n+3}}{2n+3}

: \int x^4 r\,dx = \frac{x^3r^3}{6}-\frac{a^2xr^3}{8}+\frac{a^4xr}{16}+\frac{a^6}{16}\ln\left(x+r\right)

: \int x^4 r^3\,dx = \frac{x^3r^5}{8}-\frac{a^2xr^5}{16}+\frac{a^4xr^3}{64}+\frac{3a^6xr}{128}+\frac{3a^8}{128}\ln\left(x+r\right)

: \int x^5 r\,dx = \frac{r^7}{7} - \frac{2 a^2 r^5}{5} + \frac{a^4 r^3}{3}

: \int x^5 r^3\,dx = \frac{r^9}{9} - \frac{2 a^2 r^7}{7} + \frac{a^4 r^5}{5}

: \int x^5 r^{2n+1}\,dx = \frac{r^{2n+7}}{2n+7} - \frac{2a^2r^{2n+5}}{2n+5}+\frac{a^4 r^{2n+3}}{2n+3} 

: \int\frac{r\,dx}{x} = r-a\ln\left|\frac{a+r}{x}\right| = r - a\, \operatorname{arsinh}\frac{a}{x}

: \int\frac{r^3\,dx}{x} = \frac{r^3}{3}+a^2r-a^3\ln\left|\frac{a+r}{x}\right|

: \int\frac{r^5\,dx}{x} = \frac{r^5}{5}+\frac{a^2r^3}{3}+a^4r-a^5\ln\left|\frac{a+r}{x}\right|

## ''s'' = 을 포함하는 함수의 적분
아래 적분식들에서 *x*2 > *a*2이다. (*x*2 2인 경우는 다음 목록에 있다.)

: \int s\,dx = \frac{1}{2}\left(xs-a^{2}\ln\left|x+s\right|\right)

: \int xs\,dx = \frac{1}{3}s^3

: \int\frac{s\,dx}{x} = s - |a|\arccos\left|\frac{a}{x}\right|

: \int\frac{dx}{s} = \ln\left|\frac{x+s}{a}\right|

여기서 \ln\left|\frac{x+s}{a}\right|
=\operatorname{sgn}(x)\,\operatorname{arcosh}\left|\frac{x}{a}\right|
=\frac{1}{2}\ln\left(\frac{x+s}{x-s}\right)이고, \operatorname{arcosh}\left|\frac{x}{a}\right|의 양수값을 취한다.

: \int\frac{x\,dx}{s} = s

: \int\frac{x\,dx}{s^3} = -\frac{1}{s}

: \int\frac{x\,dx}{s^5} = -\frac{1}{3s^3}

: \int\frac{x\,dx}{s^7} = -\frac{1}{5s^5}

: \int\frac{x\,dx}{s^{2n+1}} = -\frac{1}{(2n-1)s^{2n-1}} 

: \int\frac{x^{2m}\,dx}{s^{2n+1}}
= -\frac{1}{2n-1}\frac{x^{2m-1}}{s^{2n-1}}+\frac{2m-1}{2n-1}\int\frac{x^{2m-2}\,dx}{s^{2n-1}}


: \int\frac{x^2\,dx}{s}
= \frac{xs}{2}+\frac{a^2}{2}\ln\left|\frac{x+s}{a}\right|

: \int\frac{x^2\,dx}{s^3}
= -\frac{x}{s}+\ln\left|\frac{x+s}{a}\right|

: \int\frac{x^4\,dx}{s}
= \frac{x^3s}{4}+\frac{3}{8}a^2xs+\frac{3}{8}a^4\ln\left|\frac{x+s}{a}\right| 

: \int\frac{x^4\,dx}{s^3}
= \frac{xs}{2}-\frac{a^2x}{s}+\frac{3}{2}a^2\ln\left|\frac{x+s}{a}\right| 

: \int\frac{x^4\,dx}{s^5}
= -\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}+\ln\left|\frac{x+s}{a}\right| 

## ''u'' = 을 포함하는 함수의 적분

: \int u\,dx = \frac{1}{2}\left(xu+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}

: \int xu\,dx = -\frac{1}{3} u^3 \qquad\mbox{(}|x|\leq|a|\mbox{)}

: \int x^2u\,dx = -\frac{x}{4} u^3+\frac{a^2}{8}(xu+a^2\arcsin\frac{x}{a}) \qquad\mbox{(}|x|\leq|a|\mbox{)}

: \int\frac{u\,dx}{x} = u-a\ln\left|\frac{a+u}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}

: \int\frac{dx}{u} = \arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}

: \int\frac{x^2\,dx}{u} = \frac{1}{2}\left(-xu+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}

: \int u\,dx = \frac{1}{2}\left(xu-\sgn x\,\operatorname{arcosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}

: \int \frac{x}{u}\,dx = -u  \qquad\mbox{(}|x|\leq|a|\mbox{)}

## ''R'' = 를 포함하는 함수의 적분

여기서 (*ax*2 + *bx* + *c*)는 (*px* + *q*)2의 꼴로 나타낼 수 없다고 가정한다.

: \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left|2\sqrt{a}R+2ax+b\right| \qquad \mbox{(}a>0\mbox{)}

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

: \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln|2ax+b| \quad \mbox{(}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{(}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|\quad\mbox{(}c > 0\mbox{)}

: \int\frac{dx}{xR} = \frac{1}{\sqrt{-c}}\operatorname{arcsin}\left(\frac{bx+2c}}

분류:적분