Engineering Handbook/Calculus/Integration/irrational functions


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


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


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


 * $$\int x r\;dx=\frac{r^3}{3}$$


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


 * $$\int\frac{x\,dx}{r} = r$$


 * $$\int\frac{x\,dx}{r^3} = -\frac{1}{r}$$


 * $$\int\frac{x^2\;dx}{r} = \frac{x}{2}r-\frac{a^2}{2}\,\operatorname{arsinh}\frac{x}{a} = \frac{x}{2}r-\frac{a^2}{2}\ln\left( \frac {x+r}{a}  \right)$$


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

Integrals involving $$s = \sqrt{x^2-a^2}$$
Assume $$(x^2>a^2)$$, for $$(x^2<a^2)$$, see next section:


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


 * $$\int xs\;dx = \frac{1}{3}s^3$$


 * $$\int\frac{s\;dx}{x} = s - a\arccos\left|\frac{a}{x}\right|$$


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

Here $$\ln\left|\frac{x+s}{a}\right| =\mathrm{sgn}(x)\,\operatorname{arcosh}\left|\frac{x}{a}\right| =\frac{1}{2}\ln\left(\frac{x+s}{x-s}\right)$$, where the positive value of $$\operatorname{arcosh}\left|\frac{x}{a}\right|$$ is to be taken.


 * $$\int\frac{x\;dx}{s} = s$$


 * $$\int\frac{x\;dx}{s^3} = -\frac{1}{s}$$


 * $$\int\frac{x\;dx}{s^5} = -\frac{1}{3s^3}$$


 * $$\int\frac{x\;dx}{s^7} = -\frac{1}{5s^5}$$


 * $$\int\frac{x\;dx}{s^{2n+1}} = -\frac{1}{(2n-1)s^{2n-1}} $$


 * $$\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| $$


 * $$\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}{n-m-1 \choose i}\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}\qquad\mbox{(}n>m\ge0\mbox{)}$$


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


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


 * $$\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]$$


 * $$\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]$$


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


 * $$\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]$$


 * $$\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]$$

Integrals involving $$u = \sqrt{a^2-x^2}$$

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


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


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


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


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


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


 * $$\int u\;dx = \frac{1}{2}\left(xu-\sgn x\,\operatorname{arcosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge|a|\mbox{)}$$


 * $$\int \frac{x}{u}\;dx = -u \qquad\mbox{(}|x|\leq|a|\mbox{)}$$

Integrals involving $$R = \sqrt{ax^2+bx+c}$$
Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.


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


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


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


 * $$\int\frac{dx}{R} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad \mbox{(for }a<0\mbox{, }4ac-b^2<0\mbox{, }\left|2ax+b\right|<\sqrt{b^2-4ac}\mbox{)}$$


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


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


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


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


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


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


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


 * $$\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\operatorname{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right), ~ c < 0$$


 * $$\int\frac{dx}{xR}=\frac{1}{\sqrt{-c}}\operatorname{arcsin}\left(\frac{bx+2c}{|x|\sqrt{b^2-4ac}}\right), ~ c < 0, b^2-4ac>0$$


 * $$\int\frac{dx}{xR}=-\frac{2}{bx}\left(\sqrt{ax^2+bx}\right), ~ c = 0$$

Integrals involving $$S = \sqrt{ax+b}$$

 * $$\int S {dx} = \frac{2 S^{3}}{3 a}$$


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


 * $$\int \frac{dx}{x S} =

\begin{cases} -\frac{2}{\sqrt{b}} \mathrm{arcoth}\left( \frac{S}{\sqrt{b}}\right) & \mbox{(for }b > 0, \quad a x > 0\mbox{)} \\ -\frac{2}{\sqrt{b}} \mathrm{artanh}\left( \frac{S}{\sqrt{b}}\right) & \mbox{(for }b > 0, \quad a x < 0\mbox{)} \\ \frac{2}{\sqrt{-b}} \arctan\left( \frac{S}{\sqrt{-b}}\right) & \mbox{(for }b < 0\mbox{)} \\ \end{cases}$$


 * $$\int\frac{S}{x}\,dx =

\begin{cases} 2 \left( S - \sqrt{b}\,\mathrm{arcoth}\left( \frac{S}{\sqrt{b}}\right)\right) & \mbox{(for }b > 0, \quad a x > 0\mbox{)} \\ 2 \left( S - \sqrt{b}\,\mathrm{artanh}\left( \frac{S}{\sqrt{b}}\right)\right) & \mbox{(for }b > 0, \quad a x < 0\mbox{)} \\ 2 \left( S - \sqrt{-b} \arctan\left( \frac{S}{\sqrt{-b}}\right)\right) & \mbox{(for }b < 0\mbox{)} \\ \end{cases}$$


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


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


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