فهرست انتگرال تابع‌های گنگ

${\displaystyle \int rdx=\frac{1}{2}\left(xr+a^2\ln \left(x+r\right)\right)}$

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

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

${\displaystyle \int xrdx=\frac{r^3}{3}}$

${\displaystyle \int x{r}^{3}dx=\frac{r^5}{5}}$

${\displaystyle \int x{r}^{2n+1}dx=\frac{r^{2n+3}}{2n+3}}$

${\displaystyle \int x^{2}rdx=\frac{x{r}^3}{4}-\frac{a^{2}xr}{8}-\frac{a^4}{8}\ln \left(x+r\right)}$

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

${\displaystyle \int x^{3}rdx=\frac{r^5}{5}-\frac{a^2{r}^3}{3}}$

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

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

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

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

${\displaystyle \int x^{5}rdx=\frac{r^7}{7}-\frac{2a^2{r}^5}{5}+\frac{a^4{r}^3}{3}}$

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

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

${\displaystyle \int \frac{rdx}{x}=r-a\ln \left|\frac{a+r}{x}\right|=r-a\mathrm{arsinh}\frac{a}{x}}$

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

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

${\displaystyle \int \frac{r^{7}dx}{x}=\frac{r^7}{7}+\frac{a^2{r}^5}{5}+\frac{a^4{r}^3}{3}+a^{6}r-a^7\ln \left|\frac{a+r}{x}\right|}$

${\displaystyle \int \frac{dx}{r}=\mathrm{arsinh}\frac{x}{a}=\ln \left(\frac{x+r}{a}\right)}$

${\displaystyle \int \frac{dx}{r^3}=\frac{x}{a^{2}r}}$

${\displaystyle \int \frac{xdx}{r}=r}$

${\displaystyle \int \frac{xdx}{r^3}=-\frac{1}{r}}$

${\displaystyle \int \frac{x^{2}dx}{r}=\frac{x}{2}r-\frac{a^2}{2}\mathrm{arsinh}\frac{x}{a}=\frac{x}{2}r-\frac{a^2}{2}\ln \left(\frac{x+r}{a}\right)}$

${\displaystyle \int \frac{dx}{xr}=-\frac{1}{a}\mathrm{arsinh}\frac{a}{x}=-\frac{1}{a}\ln \left|\frac{a+r}{x}\right|}$

${\displaystyle \int sdx=\frac{1}{2}\left(xs-a^2\ln (x+s)\right)}$

${\displaystyle \int xsdx=\frac{1}{3}{s}^3}$

${\displaystyle \int \frac{sdx}{x}=s-a\arccos \left|\frac{a}{x}\right|}$

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

${\displaystyle \int \frac{xdx}{s}=s}$

${\displaystyle \int \frac{xdx}{s^3}=-\frac{1}{s}}$

${\displaystyle \int \frac{xdx}{s^5}=-\frac{1}{3s^3}}$

${\displaystyle \int \frac{xdx}{s^7}=-\frac{1}{5s^5}}$

${\displaystyle \int \frac{xdx}{s^{2n+1}}=-\frac{1}{(2n-1)s^{2n-1}}}$

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

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

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

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

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

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

${\displaystyle \int \frac{x^{2m}dx}{s^{2n+1}}=(-1{)}^{n-m}\frac{1}{a^{2(n-m)}}\sum _{i=0}^{n-m-1}\frac{1}{2(m+i)+1}(\genfrac{}{}{0ex}{}{n-m-1}{i})\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}\text{(}n>m\ge 0\text{)}}$

${\displaystyle \int \frac{dx}{s^3}=-\frac{1}{a^2}\frac{x}{s}}$

${\displaystyle \int \frac{dx}{s^5}=\frac{1}{a^4}\left[\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}\right]}$

${\displaystyle \int \frac{dx}{s^7}=-\frac{1}{a^6}\left[\frac{x}{s}-\frac{2}{3}\frac{x^3}{s^3}+\frac{1}{5}\frac{x^5}{s^5}\right]}$

${\displaystyle \int \frac{dx}{s^9}=\frac{1}{a^8}\left[\frac{x}{s}-\frac{3}{3}\frac{x^3}{s^3}+\frac{3}{5}\frac{x^5}{s^5}-\frac{1}{7}\frac{x^7}{s^7}\right]}$

${\displaystyle \int \frac{x^{2}dx}{s^5}=-\frac{1}{a^2}\frac{x^3}{3s^3}}$

${\displaystyle \int \frac{x^{2}dx}{s^7}=\frac{1}{a^4}\left[\frac{1}{3}\frac{x^3}{s^3}-\frac{1}{5}\frac{x^5}{s^5}\right]}$

${\displaystyle \int \frac{x^{2}dx}{s^9}=-\frac{1}{a^6}\left[\frac{1}{3}\frac{x^3}{s^3}-\frac{2}{5}\frac{x^5}{s^5}+\frac{1}{7}\frac{x^7}{s^7}\right]}$

${\displaystyle \int udx=\frac{1}{2}\left(xu+a^2\arcsin \frac{x}{a}\right)\text{(}|x|\le |a|\text{)}}$

${\displaystyle \int xudx=-\frac{1}{3}{u}^3\text{(}|x|\le |a|\text{)}}$

${\displaystyle \int x^{2}udx=-\frac{x}{4}{u}^3+\frac{a^2}{8}(xu+a^2\arcsin \frac{x}{a})\text{(}|x|\le |a|\text{)}}$

${\displaystyle \int \frac{udx}{x}=u-a\ln \left|\frac{a+u}{x}\right|\text{(}|x|\le |a|\text{)}}$

${\displaystyle \int \frac{dx}{u}=\arcsin \frac{x}{a}\text{(}|x|\le |a|\text{)}}$

${\displaystyle \int \frac{x^{2}dx}{u}=\frac{1}{2}\left(-xu+a^2\arcsin \frac{x}{a}\right)\text{(}|x|\le |a|\text{)}}$

${\displaystyle \int udx=\frac{1}{2}\left(xu-\mathrm{sgn}x\mathrm{arcosh}\left|\frac{x}{a}\right|\right)\text{(for }|x|\ge |a|\text{)}}$

${\displaystyle \int \frac{x}{u}dx=-u\text{(}|x|\le |a|\text{)}}$

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

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

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

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

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

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

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

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

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

${\displaystyle \int \frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln \left(\frac{2\sqrt{c}R+bx+2c}{x}\right),c>0}$

${\displaystyle \int \frac{dx}{xR}=-\frac{2}{bx}\left(\sqrt{a{x}^2+bx}\right),c=0}$

(برای دیدن اثبات، گزینهٔ «نمایش» را انتخاب کنید)

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ابتدا عبارت ${\displaystyle u\equiv \sqrt{ax+b}}$

را به عنوان عبارت جایگزین انتخاب می‌کنیم، بنابراین خواهیم داشت: ${\displaystyle R=\sqrt{x}u}$

، آنگاه عبارت انتگرال چنین بدست خواهد آمد:

${\displaystyle 2\sqrt{a}\int \frac{du}{(u^2-b{)}^{\frac{3}{2}}}}$

حال عبارت جایگزین دیگری مانند ${\displaystyle v\equiv \frac{1}{\sqrt{u^2-b}}}$

را انتخاب می‌کنیم، حال انتگرال خواهد شد:

${\displaystyle -2\sqrt{a}\int \frac{vdv}{\sqrt{1+b{v}^2}}}$

انتگرال به شکل ${\displaystyle \int \frac{xdx}{r}=r}$

در می‌آید، که مقدار آن مانند بالا قابل برآورد است.

البته می‌توان عبارت جانشین را به شکل ${\displaystyle u\equiv \sqrt{\frac{ax+b}{ax}}}$

، انتخاب کرد، درنتیجه:

${\displaystyle \int \frac{dx}{xR}=-\frac{2\sqrt{a}}{b}\int du=-\frac{2\sqrt{a}}{b}u}$

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${\displaystyle \int Sdx=\frac{2S^3}{3a}}$

${\displaystyle \int \frac{dx}{S}=\frac{2S}{a}}$

${\displaystyle \int \frac{x^n}{S}dx=\frac{2}{a(2n+1)}\left(x^{n}S-bn\int \frac{x^{n-1}}{S}dx\right)}$

${\displaystyle \int x^{n}Sdx=\frac{2}{a(2n+3)}\left(x^n{S}^3-nb\int x^{n-1}Sdx\right)}$

${\displaystyle \int \frac{1}{x^{n}S}dx=-\frac{1}{b(n-1)}\left(\frac{S}{x^{n-1}}+\left(n-\frac{3}{2}\right)a\int \frac{dx}{x^{n-1}S}\right)}$