Integration Formulae

Algebric Identities

Integrals

$$\int { f(x) dx } = F(x) + C $$

$$\int {dx} = x + C $$

$$\int {kf(x)dx} = k\int {f(x)dx} $$

$$\int {(a + b + c)dx} = \int {adx} + \int {bdx} + \int {cdx} $$

$$\int {n^ndx} = \frac{n^{n+1}}{n+1} + C $$

$$\int {\cos{x}dx} = \sin{x} + C $$

$$\int {\sin{x}dx} = -\cos{x} $$

$$\int {\sec^2{x}dx} = \tan{x} + C $$

$$\int {\csc^2{x}dx} = -\cot{x} + C $$

$$\int {\sec{x}\tan{x}dx} = \sec{x} + C $$

$$\int {\csc{x}\cot{x}dx} = -\csc{x} + C$$

$$\int {\frac{dx}{\sqrt{1 - x^2}}} = \sin^{-1}x + C $$

$$\int {\frac{dx}{\sqrt{1 - x^2}}} = -\cos^{-1}x + C $$

$$\int {\frac{dx}{1 + x^2}} = \tan^{-1}x + C $$

$$\int {\frac{dx}{1 + x^2}} = -\cot^{-1}x + C $$

$$\int {\frac{dx}{x\sqrt{x^2 -1}}} = \sec^{-1}x + C $$

$$\int {\frac{dx}{x\sqrt{x^2 -1}}} = -\csc^{-1}x + C $$

$$\int {\frac{dx}{x^2 - a^2}} = \frac{1}{2a}\log{\left|\frac{x-a}{x + a}\right|} + C $$

$$\int {\frac{dx}{a^2 - x^2}} = \frac{1}{2a}\log{\left|\frac{a + x}{a - x}\right|} + C $$

$$\int {\frac{dx}{x^2 + a^2}} = \frac{1}{a}\tan^{-1}({\frac{x}{a})} + C $$

$$\int {\frac{dx}{\sqrt{x^2 + a^2}}} = \log{\left|x + \sqrt{x^2 + a^2} \right|} + C $$

$$\int {\frac{dx}{\sqrt{a^2 - x^2}}} = \sin^{-1}{(\frac{x}{a})} + C $$

$$\int {\frac{dx}{\sqrt{x^2 + a^2}}} = \log{\left|x + \sqrt{x^2 - a^2} \right|} + C $$

$$\int {\tan{x}dx} = \log{\left|\sec{x}\right|} + C$$

$$\int{\cot{x}dx} = \log{\left|\sin{x}\right|} + C $$

$$\int{\sec{x}dx} = \log{\left|\sec{x} + \tan{x}\right|} + C $$

$$\int{\csc{x}dx} = \log{\left|\csc{x} - \cot{x}\right|} + C $$

$$\int{e^x}dx = e^x + C$$

$$\int{a^x}dx = \frac{a^x}{\log{a}} + C $$