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Table of Indefinite Integrals


This table of indefinite integrals is by no means an exhautive list, but it includes most of the forms you might encounter regularly, as well as a few more obscure ones.


(1)\(\displaystyle\int x^n\,dx\) \(\displaystyle=\frac{x^{n+1}}{n+1}+c\), \((n \ne -1)\)

(2)\(\displaystyle\int x^{-1}\,dx\) \(\displaystyle=ln\,x+c\)

(3)\(\displaystyle\int a^x\,dx\) \(\displaystyle=a^x\,log_a\,ε+c\)

(4)\(\displaystyle\int\frac{dx}{x^2+a^2}\) \(\displaystyle=\frac{1}{a}tan^{-1}\,\frac{x}{a}+c\)

(5)\(\displaystyle\int\frac{dx}{x^2-a^2}\) \(\displaystyle=\frac{1}{2a}\,ln\frac{x-a}{x+a}+c\), if \((x^2>a^2)\) or \(\frac{1}{2a}\,ln\frac{a-x}{a+x}+c\), if \((x^2<a^2)\)

(6)\(\displaystyle\int\frac{dx}{\sqrt{a^2-x^2}}\) \(\displaystyle=sin^{-1}\frac{x}{a}+c\)

(7)\(\displaystyle\int\frac{dx}{\sqrt{x^2\pm a^2}}\) \(\displaystyle=ln(x+\sqrt{x^2\pm a^2})+c\)

(8)\(\displaystyle\int\sqrt{a^2-x^2}\,dx\) \(\displaystyle\displaystyle=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}+c\)

(9)\(\displaystyle\int x\sqrt{a^2-x^2}\,dx\) \(\displaystyle=-\frac{1}{3}(a^2-x^2)^{\frac{3}{2}}+c\)

(10)\(\displaystyle\int x^2\sqrt{a^2-x^2}\,dx\) \(\displaystyle=\frac{x}{8}(2x^2)\sqrt{a^2-x^2}+\frac{a^4}{8}sin^{-1}\frac{x}{a}+c\)

(11)\(\displaystyle\int \frac{x\,dx}{\sqrt{a^2-x^2}}\) \(\displaystyle=-\sqrt{a^2-x^2}+c\)

(12)\(\displaystyle\int \frac{x^2\,dx}{\sqrt{a^2-x^2}}\) \(\displaystyle=-\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}+c\)

(13)\(\displaystyle\int(a^2-x^2)^{\frac{3}{2}}\,dx\) \(\displaystyle=\frac{x}{8}(5a^2-2x^2)\sqrt{a^2-x^2}+\frac{3a^4}{8}sin^{-1}\frac{x}{a}+c\)

(14)\(\displaystyle\int\frac{dx}{(a^2-x^2)^{\frac{3}{2}}}\) \(\displaystyle=\frac{x}{a^2\sqrt{a^2-x^2}}+c\)

(15)\(\displaystyle\int\frac{x\,dx}{(a^2-x^2)^{\frac{3}{2}}}\) \(\displaystyle=\frac{1}{\sqrt{a^2-x^2}}+c\)

(16)\(\displaystyle\int\frac{x^2\,dx}{(a^2-x^2)^{\frac{3}{2}}}\) \(\displaystyle=\frac{x}{\sqrt{a^2-x^2}}-sin^{-1}\frac{x}{a}+c\)

(17)\(\displaystyle\int\) \(=\)

(18)\(\displaystyle\int\) \(=\)

(19)\(\displaystyle\int\) \(=\)

(20)\(\displaystyle\int\) \(=\)