List of integrals of irrational functions

The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.

Integrals involving ''r'' =

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

: $\int r^3\,dx = \fracxr^3+\fraca^2xr+\fraca^4\ln\left(x+r\right)$

: $\int r^5\,dx = \fracxr^5+\fraca^2xr^3+\fraca^4xr+\fraca^6\ln\left(x+r\right)$

: $\int x r\,dx = \frac$

: $\int x r^3\,dx = \frac$

: $\int x r^\,dx = \frac$

: $\int x^2 r\,dx = \frac-\frac-\frac\ln\left(x+r\right)$

: $\int x^2 r^3\,dx = \frac-\frac-\frac-\frac\ln\left(x+r\right)$

: $\int x^3 r\,dx = \frac - \frac$

: $\int x^3 r^3\,dx = \frac-\frac$

: $\int x^3 r^\,dx = \frac - \frac$

: $\int x^4 r\,dx = \frac-\frac+\frac+\frac\ln\left(x+r\right)$

: $\int x^4 r^3\,dx = \frac-\frac+\frac+\frac+\frac\ln\left(x+r\right)$

: $\int x^5 r\,dx = \frac - \frac + \frac$

: $\int x^5 r^3\,dx = \frac - \frac + \frac$

: $\int x^5 r^\,dx = \frac - \frac+\frac$

: $\int\frac = r-a\ln\left, \frac\ = r - a\, \operatorname\frac$

: $\int\frac = \frac+a^2r-a^3\ln\left, \frac\$

: $\int\frac = \frac+\frac+a^4r-a^5\ln\left, \frac\$

: $\int\frac = \frac+\frac+\frac+a^6r-a^7\ln\left, \frac\$

: $\int\frac = \operatorname\frac = \ln\left( \frac \right)$

: $\int\frac = \frac$

: $\int\frac = r$

: $\int\frac = -\frac$

: $\int\frac = \fracr-\frac\,\operatorname\frac = \fracr-\frac\ln\left( \frac \right)$

: $\int\frac = -\frac\,\operatorname\frac = -\frac\ln\left, \frac\$

Integrals involving ''s'' =

Assume ''x''² > ''a''² (for ''x''² < ''a''², see next section):

: $\int s\,dx = \frac\left(xs-a^\ln\left, x+s\\right)$

: $\int xs\,dx = \fracs^3$

: $\int\frac = s - , a, \arccos\left, \frac\$

: $\int\frac = \ln\left, \frac\$
Here $\ln\left, \frac\ =\operatorname(x)\,\operatorname\left, \frac\ =\frac\ln\left(\frac\right)$, where the positive value of $\operatorname\left, \frac\$ is to be taken.

: $\int\frac = s$

: $\int\frac = -\frac$

: $\int\frac = -\frac$

: $\int\frac = -\frac$

: $\int\frac = -\frac$

: $\int\frac = -\frac\frac+\frac\int\frac$

: $\int\frac = \frac+\frac\ln\left, \frac\$

: $\int\frac = -\frac+\ln\left, \frac\$

: $\int\frac = \frac+\fraca^2xs+\fraca^4\ln\left, \frac\$

: $\int\frac = \frac-\frac+\fraca^2\ln\left, \frac\$

: $\int\frac = -\frac-\frac\frac+\ln\left, \frac\$

: $\int\frac = (-1)^\frac\sum_^\frac\frac\qquad\mboxn>m\ge0\mbox$

: $\int\frac = -\frac\frac$

: \int\frac = \frac\left frac-\frac\frac\right/math>

: \int\frac
=-\frac\left frac-\frac\frac+\frac\frac\right/math>

: \int\frac
=\frac\left frac-\frac\frac+\frac\frac-\frac\frac\right/math>

: $\int\frac = -\frac\frac$

: \int\frac
= \frac\left frac\frac-\frac\frac\right/math>

: \int\frac
= -\frac\left frac\frac-\frac\frac+\frac\frac\right/math>

Integrals involving ''u'' =

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

: $\int xu\,dx = -\frac u^3 \qquad\mbox, x, \leq, a, \mbox$

: $\int x^2u\,dx = -\frac u^3+\frac(xu+a^2\arcsin\frac) \qquad\mbox, x, \leq, a, \mbox$

: $\int\frac = u-a\ln\left, \frac\ \qquad\mbox, x, \leq, a, \mbox$

: $\int\frac = \arcsin\frac \qquad\mbox, x, \leq, a, \mbox$

: $\int\frac = \frac\left(-xu+a^2\arcsin\frac\right) \qquad\mbox, x, \leq, a, \mbox$

: $\int u\,dx = \frac\left(xu-\sgn x\,\operatorname\left, \frac\\right) \qquad\mbox, x, \ge, a, \mbox$

:$\int \frac\,dx = -u \qquad\mbox, x, \leq, a, \mbox$

Integrals involving ''R'' =

Assume (''ax''² + ''bx'' + ''c'') cannot be reduced to the following expression (''px'' + ''q'')² for some ''p'' and ''q''.

: $\int\frac = \frac\ln\left, 2\sqrtR+2ax+b\ \qquad \mboxa>0\mbox$

: $\int\frac = \frac\,\operatorname\frac \qquad \mboxa>0\mbox4ac-b^2>0\mbox$

: $\int\frac = \frac\ln, 2ax+b, \quad \mboxa>0\mbox4ac-b^2=0\mbox$

: $\int\frac = -\frac\arcsin\frac \qquad \mboxa<0\mbox4ac-b^2<0\mbox\left, 2ax+b\<\sqrt\mbox$

: $\int\frac = \frac$

: $\int\frac = \frac\left(\frac+\frac\right)$

: $\int\frac = \frac\left(\frac+4a(n-1)\int\frac\right)$

: $\int\frac\,dx = \frac-\frac\int\frac$

: $\int\frac\,dx = -\frac$

: $\int\frac\,dx = -\frac-\frac\int\frac$

: $\int\frac = -\frac\ln \left, \frac\, ~ c > 0$

: $\int\frac = -\frac\operatorname\left(\frac\right), ~ c < 0$

: $\int\frac = \frac\operatorname\left(\frac\right), ~ c < 0, b^2-4ac>0$

: $\int\frac = -\frac\left(\sqrt\right), ~ c = 0$

: $\int\frac\,dx = \fracR+\frac\int\frac$

: $\int \frac = -\frac-\frac \int \frac$

: $\int R\,dx = \frac R + \frac \int \frac$

: $\int x R\,dx = \frac-\frac R - \frac \int \frac$

: $\int x^ R\,dx = \fracR^3+\frac \int R\,dx$

: $\int \frac\,dx = R + \frac \int \frac+c \int \frac$

: $\int \frac\,dx = -\frac+a \int \frac+ \frac \int \frac$

: $\int \frac = \frac+ \frac \int \frac$

Integrals involving ''S'' =

: $\int S\,dx = \frac$

: $\int \frac = \frac$

: $\int \frac = \begin -\dfrac \operatorname\left( \dfrac\right) & \mboxb > 0, \quad a x > 0\mbox \\ -\dfrac \operatorname\left( \dfrac\right) & \mboxb > 0, \quad a x < 0\mbox \\ \dfrac \arctan\left( \dfrac\right) & \mboxb < 0\mbox \\ \end$

: $\int\frac\,dx = \begin 2 \left( S - \sqrt\,\operatorname\left( \dfrac\right)\right) & \mboxb > 0, \quad a x > 0\mbox \\ 2 \left( S - \sqrt\,\operatorname\left( \dfrac\right)\right) & \mboxb > 0, \quad a x < 0\mbox \\ 2 \left( S - \sqrt \arctan\left( \dfrac\right)\right) & \mboxb < 0\mbox \\ \end$

: $\int \frac\,dx = \frac \left( x^ S - b n \int \frac\,dx\right)$

: $\int x^ S\,dx = \frac \left(x^ S^ - n b \int x^ S\,dx\right)$

: $\int \frac\,dx = -\frac \left( \frac + \left( n - \frac\right) a \int \frac\right)$

References

*
* Milton Abramowitz and Irene A. Stegun, eds., '' Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables'' 1972, Dover: New York. ''(Se
chapter 3
)''
* (Several previous editions as well.)
{{Lists of integrals

Irrational functions