İrrasyonel fonksiyonların integralleri
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Bu irrasyonel fonksiyonların integrallerini (ters türevlerini) barındıran bir listedir. Farklı fonksiyonların integrallerine ait bilgi için integral tablosu sayfasına göz atabilirsiniz. \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{1}{8}3a^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^3 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 \sinh^{-1}\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 \int\frac{r^7\;dx}{x} = \frac{r^7}{7}+\frac{a^2r^5}{5}+\frac{a^4r^3}{3}+a^6r-a^7\ln\left \frac{a+r}{x}\right \int\frac{dx}{r} = \sinh^{-1}\frac{x}{a} = \ln\left x+r\right \int\frac{dx}{r^3} = \frac{x}{a^2r} \int\frac{x\,dx}{r} = r \int\frac{x\,dx}{r^3} = -\frac{1}{r} \int\frac{x^2\;dx}{r} = \frac{x}{2}r-\frac{a^2}{2}\,\sinh^{-1}\frac{x}{a} = \frac{x}{2}r-\frac{a^2}{2}\ln\left x+r\right \int\frac{dx}{xr} = -\frac{1}{a}\,\sinh^{-1}\frac{a}{x} = -\frac{1}{a}\ln\left \frac{a+r}{x}\right (x^2>a^2) olduğunu varsayın.