무리함수 적분표

아래 목록은 무리 함수의 부정적분이다. 아래의 적분식들에서 적분상수는 생략하였다. == *r* = *a*2 \+ *x*2을 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* = *x*2 − *a*2을 포함하는 함수의 적분 == 아래 적분식들에서 *x*2 > *a*2이다. (*x*2 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 0\mbox{)} : \int\frac{dx}{xR} = \frac{1}{\sqrt{-c\operatorname{arcsin}\left(\frac{bx+2c}{|x|\sqrt{b^2-4ac\right)\quad\mbox{(}c 0\mbox{)} : \int\frac{dx}{xR} = -\frac{2}{bx}\left(\sqrt{ax^2+bx}\right)\quad\mbox{(}c = 0\mbox{)} : \int\frac{x^2}{R}\,dx = \frac{2ax-3b}{4a^2}R+\frac{3b^2-4ac}{8a^2}\int\frac{dx}{R} : \int \frac{dx}{x^{2} R} = -\frac{ R}{cx}-\frac{b}{2c} \int \frac{dx}{x R} : \int R\,dx = \frac{2ax+b}{4a} R + \frac{4ac-b^{2{8a} \int \frac{dx}{ R} : \int x R\,dx = \frac{R^3}{3a}-\frac{b(2ax+b)}{8a^{2 R - \frac{b(4ac-b^{2})}{16a^{2 \int \frac{dx}{ R} : \int x^{2} R\,dx = \frac{6ax-5b}{24a^{2R^3+\frac{5b^{2}-4ac}{16a^{2 \int R\,dx : \int \frac{ R}{x}\,dx = R + \frac{b}{2} \int \frac{dx}{ R}+c \int \frac{dx}{x R} : \int \frac{ R}{x^{2\,dx = -\frac{ R}{x}+a \int \frac{dx}{R}+ \frac{b}{2} \int \frac{dx}{ xR} : \int \frac{x^{2}\,dx}{R^3} = \frac{(2b^{2}-4ac)x+2bc}{a(4ac-b^{2}) R}+ \frac{1}{a} \int \frac{dx}{ R} == *S* = *ax* + *b*를 포함하는 함수의 적분 == : \int S\,dx = \frac{2 S^{3{3 a} : \int \frac{dx}{S} = \frac{2S}{a} : \int \frac{dx}{x S} = \begin{cases} -\dfrac{2}{\sqrt{b \operatorname{arcoth}\left( \dfrac{S}{\sqrt{b\right) & \mbox{(}b > 0, \quad a x > 0\mbox{)} \\\ -\dfrac{2}{\sqrt{b \operatorname{artanh}\left( \dfrac{S}{\sqrt{b\right) & \mbox{(}b > 0, \quad a x 0, \quad a x > 0\mbox{)} \\\ 2 \left( S - \sqrt{b}\,\operatorname{artanh}\left( \dfrac{S}{\sqrt{b\right)\right) & \mbox{(}b > 0, \quad a x < 0\mbox{)} \\\ 2 \left( S - \sqrt{-b} \arctan\left( \dfrac{S}{\sqrt{-b\right)\right) & \mbox{(}b < 0\mbox{)} \\\ \end{cases} : \int \frac{x^{n{S}\,dx = \frac{2}{a (2 n + 1)} \left( x^{n} S - b n \int \frac{x^{n - 1{S}\,dx\right) : \int x^{n} S\,dx = \frac{2}{a (2 n + 3)} \left(x^{n} S^{3} - n b \int x^{n - 1} S\,dx\right) : \int \frac{1}{x^{n} S}\,dx = -\frac{1}{b (n - 1)} \left( \frac{S}{x^{n - 1 + \left( n - \frac{3}{2}\right) a \int \frac{dx}{x^{n - 1} S}\right)