Indefinite Integration

Basic Formulas

1. Power Rule

$(n\ne -1)$

${\displaystyle \int (f(x){)}^n{f}^{\prime }(x)dx={\displaystyle \frac{(f(x){)}^{n+1}}{n+1}}+C}$

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

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

2. Logarithmic Integration

${\displaystyle \int {\displaystyle \frac{f^{\prime }(x)}{f(x)}}dx=\ln |f(x)|+C}$

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

3. Trigonometric Functions

$\begin{array}{rlrl}{\displaystyle \int \sin xdx}& & & =-\cos x+C\\ {\displaystyle \int \cos xdx}& & & =\sin x+C\\ {\displaystyle \int \tan xdx}& & & =-\ln |\cos x|+C& & =\ln |\sec x|+C\\ {\displaystyle \int \cot xdx}& & & =\ln |\sin x|+C& & =-\ln |\csc x|+C\\ {\displaystyle \int \sec xdx}& & & =\ln |\sec x+\tan x|+C& & =\ln |\tan (\frac{\pi }{4}+\frac{x}{2})|+C\\ {\displaystyle \int \csc xdx}& & & =\ln |\csc x-\cot x|+C& & =\ln |\tan \frac{x}{2}|+C\end{array}$

4. Exponential Function

${\displaystyle \int e^x(f(x)+f^{\prime }(x))dx=e^{x}f(x)+C}$

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

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

5. Special

$\begin{array}{rl}{\displaystyle \int {\displaystyle \frac{dx}{a^2+x^2}}}& ={\displaystyle \frac{1}{a}}{\tan }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int {\displaystyle \frac{dx}{a^2-x^2}}}& ={\displaystyle \frac{1}{2a}}\ln \left|{\displaystyle \frac{a+x}{a-x}}\right|+C\\ {\displaystyle \int {\displaystyle \frac{dx}{\sqrt{a^2-x^2}}}}& ={\sin }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int {\displaystyle \frac{dx}{\sqrt{x^2+a^2}}}}& =\ln |x+\sqrt{x^2+a^2}|+C\\ {\displaystyle \int {\displaystyle \frac{dx}{\sqrt{x^2-a^2}}}}& =\ln |x+\sqrt{x^2-a^2}|+C\\ {\displaystyle \int {\displaystyle \frac{dx}{x\sqrt{x^2-a^2}}}}& ={\displaystyle \frac{1}{a}}{\sec }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int \sqrt{a^2-x^2}dx}& ={\displaystyle \frac{x}{2}}\sqrt{a^2-x^2}+{\displaystyle \frac{a^2}{2}}{\sin }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int \sqrt{x^2+a^2}dx}& ={\displaystyle \frac{x}{2}}\sqrt{x^2+a^2}+{\displaystyle \frac{a^2}{2}}\ln |x+\sqrt{x^2+a^2}|+C\\ {\displaystyle \int \sqrt{x^2-a^2}dx}& ={\displaystyle \frac{x}{2}}\sqrt{x^2-a^2}-{\displaystyle \frac{a^2}{2}}\ln |x+\sqrt{x^2-a^2}|+C\end{array}$