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 $r\; =\; sqrt\{x^2+a^2\}$ =

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

: $intfrac\{x,dx\}\{r\}\; =\; r$

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

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

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

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

Assume $(x^2>a^2)$, for

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

: $intfrac\{dx\}\{s\}\; =\; intfrac\{dx\}\{sqrt\{x^2-a^2\; =lnleft|frac\{x+s\}\{a\}\; ight|$Note that $lnleft|frac\{x+s\}\{a\}\; ight$
=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 $operatorname\{arcosh\}left|frac\{x\}\{a\}\; ight|$ is to be taken.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

:

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

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

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

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

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

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

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

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

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

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

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

:

:

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

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

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

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.)