# 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)$$\displaystyle\int$$ $$=$$