# List of integrals of irrational functions

List of integrals of irrational functions

__NOTOC__The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals.

= Integrals involving $r = sqrt\left\{x^2+a^2\right\}$ =

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

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

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

: $int x r;dx=frac\left\{r^3\right\}\left\{3\right\}$

: $int x r^3;dx=frac\left\{r^5\right\}\left\{5\right\}$

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

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

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

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

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

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

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

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

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

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

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

: $intfrac\left\{r;dx\right\}\left\{x\right\} = r-alnleft|frac\left\{a+r\right\}\left\{x\right\} ight| = r - a, operatorname\left\{arsinh\right\}frac\left\{a\right\}\left\{x\right\}$

: $intfrac\left\{r^3;dx\right\}\left\{x\right\} = frac\left\{r^3\right\}\left\{3\right\}+a^2r-a^3lnleft|frac\left\{a+r\right\}\left\{x\right\} ight|$

: $intfrac\left\{r^5;dx\right\}\left\{x\right\} = frac\left\{r^5\right\}\left\{5\right\}+frac\left\{a^2r^3\right\}\left\{3\right\}+a^4r-a^5lnleft|frac\left\{a+r\right\}\left\{x\right\} ight|$

: $intfrac\left\{r^7;dx\right\}\left\{x\right\} = frac\left\{r^7\right\}\left\{7\right\}+frac\left\{a^2r^5\right\}\left\{5\right\}+frac\left\{a^4r^3\right\}\left\{3\right\}+a^6r-a^7lnleft|frac\left\{a+r\right\}\left\{x\right\} ight|$

: $intfrac\left\{dx\right\}\left\{r\right\} = operatorname\left\{arsinh\right\}frac\left\{x\right\}\left\{a\right\} = lnleft|x+r ight|$

: $intfrac\left\{dx\right\}\left\{r^3\right\} = frac\left\{x\right\}\left\{a^2r\right\}$

: $intfrac\left\{x,dx\right\}\left\{r\right\} = r$

: $intfrac\left\{x,dx\right\}\left\{r^3\right\} = -frac\left\{1\right\}\left\{r\right\}$

: $intfrac\left\{x^2;dx\right\}\left\{r\right\} = frac\left\{x\right\}\left\{2\right\}r-frac\left\{a^2\right\}\left\{2\right\},operatorname\left\{arsinh\right\}frac\left\{x\right\}\left\{a\right\} = frac\left\{x\right\}\left\{2\right\}r-frac\left\{a^2\right\}\left\{2\right\}lnleft|x+r ight|$

: $intfrac\left\{dx\right\}\left\{xr\right\} = -frac\left\{1\right\}\left\{a\right\},operatorname\left\{arsinh\right\}frac\left\{a\right\}\left\{x\right\} = -frac\left\{1\right\}\left\{a\right\}lnleft|frac\left\{a+r\right\}\left\{x\right\} ight|$

= Integrals involving $s = sqrt\left\{x^2-a^2\right\}$=

Assume $\left(x^2>a^2\right)$, for

: $intfrac\left\{s;dx\right\}\left\{x\right\} = s - aarccosleft|frac\left\{a\right\}\left\{x\right\} ight|$

: $intfrac\left\{dx\right\}\left\{s\right\} = intfrac\left\{dx\right\}\left\{sqrt\left\{x^2-a^2 =lnleft|frac\left\{x+s\right\}\left\{a\right\} ight|$Note that $lnleft|frac\left\{x+s\right\}\left\{a\right\} ight$
=mathrm{sgn}(x),operatorname{arcosh}left|frac{x}{a} ight
=frac{1}{2}lnleft(frac{x+s}{x-s} ight), where the positive value of $operatorname\left\{arcosh\right\}left|frac\left\{x\right\}\left\{a\right\} ight|$ is to be taken.

: $intfrac\left\{x;dx\right\}\left\{s\right\} = s$

: $intfrac\left\{x;dx\right\}\left\{s^3\right\} = -frac\left\{1\right\}\left\{s\right\}$

: $intfrac\left\{x;dx\right\}\left\{s^5\right\} = -frac\left\{1\right\}\left\{3s^3\right\}$

: $intfrac\left\{x;dx\right\}\left\{s^7\right\} = -frac\left\{1\right\}\left\{5s^5\right\}$

: $intfrac\left\{x;dx\right\}\left\{s^\left\{2n+1 = -frac\left\{1\right\}\left\{\left(2n-1\right)s^\left\{2n-1$

: $intfrac\left\{x^\left\{2m\right\};dx\right\}\left\{s^\left\{2n+1= -frac\left\{1\right\}\left\{2n-1\right\}frac\left\{x^\left\{2m-1\left\{s^\left\{2n-1+frac\left\{2m-1\right\}\left\{2n-1\right\}intfrac\left\{x^\left\{2m-2\right\};dx\right\}\left\{s^\left\{2n-1$

: $intfrac\left\{x^2;dx\right\}\left\{s\right\}= frac\left\{xs\right\}\left\{2\right\}+frac\left\{a^2\right\}\left\{2\right\}lnleft|frac\left\{x+s\right\}\left\{a\right\} ight|$

: $intfrac\left\{x^2;dx\right\}\left\{s^3\right\}= -frac\left\{x\right\}\left\{s\right\}+lnleft|frac\left\{x+s\right\}\left\{a\right\} ight|$

: $intfrac\left\{x^4;dx\right\}\left\{s\right\}= frac\left\{x^3s\right\}\left\{4\right\}+frac\left\{3\right\}\left\{8\right\}a^2xs+frac\left\{3\right\}\left\{8\right\}a^4lnleft|frac\left\{x+s\right\}\left\{a\right\} ight|$

: $intfrac\left\{x^4;dx\right\}\left\{s^3\right\}= frac\left\{xs\right\}\left\{2\right\}-frac\left\{a^2x\right\}\left\{s\right\}+frac\left\{3\right\}\left\{2\right\}a^2lnleft|frac\left\{x+s\right\}\left\{a\right\} ight|$

: $intfrac\left\{x^4;dx\right\}\left\{s^5\right\}= -frac\left\{x\right\}\left\{s\right\}-frac\left\{1\right\}\left\{3\right\}frac\left\{x^3\right\}\left\{s^3\right\}+lnleft|frac\left\{x+s\right\}\left\{a\right\} ight|$

: $intfrac\left\{x^\left\{2m\right\};dx\right\}\left\{s^\left\{2n+1= \left(-1\right)^\left\{n-m\right\}frac\left\{1\right\}\left\{a^\left\{2\left(n-m\right)sum_\left\{i=0\right\}^\left\{n-m-1\right\}frac\left\{1\right\}\left\{2\left(m+i\right)+1\right\}\left\{n-m-1 choose i\right\}frac\left\{x^\left\{2\left(m+i\right)+1\left\{s^\left\{2\left(m+i\right)+1qquadmbox\left\{\left(\right\}n>mge0mbox\left\{\right)\right\}$

: $intfrac\left\{dx\right\}\left\{s^3\right\}=-frac\left\{1\right\}\left\{a^2\right\}frac\left\{x\right\}\left\{s\right\}$

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

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

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

: $intfrac\left\{x^2;dx\right\}\left\{s^5\right\}=-frac\left\{1\right\}\left\{a^2\right\}frac\left\{x^3\right\}\left\{3s^3\right\}$

: $intfrac\left\{x^2;dx\right\}\left\{s^7\right\}= frac\left\{1\right\}\left\{a^4\right\}left \left[frac\left\{1\right\}\left\{3\right\}frac\left\{x^3\right\}\left\{s^3\right\}-frac\left\{1\right\}\left\{5\right\}frac\left\{x^5\right\}\left\{s^5\right\} ight\right]$

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

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

: $int u ;dx = frac\left\{1\right\}\left\{2\right\}left\left(xu+a^2arcsinfrac\left\{x\right\}\left\{a\right\} ight\right) qquadmbox\left\{\left(\right\}|x|leq|a|mbox\left\{\right)\right\}$

: $int xu;dx = -frac\left\{1\right\}\left\{3\right\} u^3 qquadmbox\left\{\left(\right\}|x|leq|a|mbox\left\{\right)\right\}$

: $intfrac\left\{u;dx\right\}\left\{x\right\} = u-alnleft|frac\left\{a+u\right\}\left\{x\right\} ight| qquadmbox\left\{\left(\right\}|x|leq|a|mbox\left\{\right)\right\}$

: $intfrac\left\{dx\right\}\left\{u\right\} = arcsinfrac\left\{x\right\}\left\{a\right\} qquadmbox\left\{\left(\right\}|x|leq|a|mbox\left\{\right)\right\}$

: $intfrac\left\{x^2;dx\right\}\left\{u\right\} = frac\left\{1\right\}\left\{2\right\}left\left(-xu+a^2arcsinfrac\left\{x\right\}\left\{a\right\} ight\right) qquadmbox\left\{\left(\right\}|x|leq|a|mbox\left\{\right)\right\}$

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

= Integrals involving $R = sqrt\left\{ax^2+bx+c\right\}$=

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

: $intfrac\left\{dx\right\}\left\{R\right\} = frac\left\{1\right\}\left\{sqrt\left\{alnleft|2sqrt\left\{a\right\}R+2ax+b ight| qquad mbox\left\{\left(for \right\}a>0mbox\left\{\right)\right\}$

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

: $intfrac\left\{dx\right\}\left\{R\right\} = frac\left\{1\right\}\left\{sqrt\left\{aln|2ax+b| quad mbox\left\{\left(for \right\}a>0mbox\left\{, \right\}4ac-b^2=0mbox\left\{\right)\right\}$

:

: $intfrac\left\{dx\right\}\left\{R^3\right\} = frac\left\{4ax+2b\right\}\left\{\left(4ac-b^2\right)R\right\}$

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

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

: $intfrac\left\{x\right\}\left\{R\right\};dx = frac\left\{R\right\}\left\{a\right\}-frac\left\{b\right\}\left\{2a\right\}intfrac\left\{dx\right\}\left\{R\right\}$

: $intfrac\left\{x\right\}\left\{R^3\right\};dx = -frac\left\{2bx+4c\right\}\left\{\left(4ac-b^2\right)R\right\}$

: $intfrac\left\{x\right\}\left\{R^\left\{2n+1;dx = -frac\left\{1\right\}\left\{\left(2n-1\right)aR^\left\{2n-1-frac\left\{b\right\}\left\{2a\right\}intfrac\left\{dx\right\}\left\{R^\left\{2n+1$

: $intfrac\left\{dx\right\}\left\{xR\right\}=-frac\left\{1\right\}\left\{sqrt\left\{clnleft\left(frac\left\{2sqrt\left\{c\right\}R+bx+2c\right\}\left\{x\right\} ight\right)$

: $intfrac\left\{dx\right\}\left\{xR\right\}=-frac\left\{1\right\}\left\{sqrt\left\{coperatorname\left\{arsinh\right\}left\left(frac\left\{bx+2c\right\}\left\{|x|sqrt\left\{4ac-b^2 ight\right)$

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

: $int S \left\{dx\right\} = frac\left\{2 S^\left\{3\left\{3 a\right\}$

: $int frac\left\{dx\right\}\left\{S\right\} = frac\left\{2S\right\}\left\{a\right\}$

:

:

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

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

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

References

* cite book
last=Peirce
first=Benjamin Osgood
title=A Short Table of Integrals
origyear=1899
edition=3rd revised ed.
year=1929
publisher=Ginn and Co.
location=Boston
pages=pp. 16-30
chapter=Chap. 3

* Milton Abramowitz and Irene A. Stegun, eds., "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables" 1972, Dover: New York. "(See [http://www.math.sfu.ca/~cbm/aands/page_13.htm chapter 3] .)"

* S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)

Wikimedia Foundation. 2010.

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