Indefinite Integration

Basic Formulas

1. Power Rule

$\quad (n\not ={-1})$

$\displaystyle\int (f(x))^n f'(x) \, dx = \dfrac{(f(x))^{n+1}}{n+1} + C$

$\displaystyle\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C$

$\displaystyle\int |x|^n \, dx = \dfrac{x |x|^{n}}{n+1} + C$

2. Logarithmic Integration

$\displaystyle\int \dfrac{f'(x)}{f(x)} \, dx = \ln |f(x)| + C$

$\displaystyle\int \dfrac{1}{x} \, dx = \ln |x| + C$

3. Trigonometric Functions

$\begin{aligned} \displaystyle\int \sin x \, dx &&& = -\cos x + C \\ \displaystyle\int \cos x \, dx &&& = \sin x + C \\ \displaystyle\int \tan x \, dx &&& = -\ln |\cos x| + C && = \ln |\sec x| + C \\ \displaystyle\int \cot x \, dx &&& = \ln |\sin x| + C && = -\ln |\csc x| + C \\ \displaystyle\int \sec x \, dx &&& = \ln |\sec x + \tan x| + C && = \ln \left| \tan \left(\frac{\pi}{4} + \frac{x}{2}\right) \right| + C \\ \displaystyle\int \csc x \, dx &&& = \ln |\csc x - \cot x| + C && = \ln \left| \tan \frac{x}{2} \right| + C \end{aligned}$

4. Exponential Function

$\displaystyle\int a^x \, dx = \dfrac{a^x}{\ln a} + C$

$\displaystyle\int e^x \, dx = e^x + C$

5. Special

$\begin{aligned} \displaystyle\int \dfrac{dx}{a^2 + x^2} & = \dfrac{1}{a} \tan^{-1} \dfrac{x}{a} + C \\[8mu] \displaystyle\int \dfrac{dx}{a^2 - x^2} & = \dfrac{1}{2a} \ln \left| \dfrac{a+x}{a-x} \right| + C \\[25mu] \displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}} & = \sin^{-1} \dfrac{x}{a} + C \\[25mu] \displaystyle\int \dfrac{dx}{\sqrt{x^2 + a^2}} & = \ln \left|x + \sqrt{x^2 + a^2}\right| + C \\[8mu] \displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}} & = \ln \left| x + \sqrt{x^2 - a^2} \right| + C \\[25mu] \displaystyle\int \dfrac{dx}{x\sqrt{x^2 - a^2}} & = \dfrac{1}{a} \sec^{-1} \dfrac{x}{a} + C \\[25mu] \displaystyle\int \sqrt{a^2 - x^2} \, dx & = \dfrac{x}{2} \sqrt{a^2 - x^2} + \dfrac{a^2}{2} \sin^{-1} \dfrac{x}{a} + C \\[8mu] \displaystyle\int \sqrt{x^2 + a^2} \, dx & = \dfrac{x}{2} \sqrt{x^2 + a^2} + \dfrac{a^2}{2} \ln \left|x + \sqrt{x^2 + a^2}\right| + C \\[8mu] \displaystyle\int \sqrt{x^2 - a^2} \, dx & = \dfrac{x}{2} \sqrt{x^2 - a^2} - \dfrac{a^2}{2} \ln \left|x + \sqrt{x^2 - a^2}\right| + C \\[8mu] \end{aligned}$