Basic Integration

Elementary Forms

\[\begin{split} &\displaystyle{\int e^x \; dx} = e^x +C & &\displaystyle{\int \dfrac{1}{x} \; dx} = \ln |x|+C \\ &\displaystyle{\int \sin x \; dx} = -\cos x +C & &\displaystyle{\int \cos x \; dx} = \sin x+C \\ &\displaystyle{\int \sec^2 x \; dx} = \tan x +C & &\displaystyle{\int \csc^2 x \; dx} = -\cot x+C \\ &\displaystyle{\int \sec x \tan x \; dx} = \sec x +C \qquad & &\displaystyle{\int \csc x \cot x \; dx} = -\csc x+C \\ &\displaystyle{\int \dfrac{1}{1+x^2} \; dx} = \tan^{-1} x +C & &\displaystyle{\int \dfrac{1}{\sqrt{1-x^2}} \; dx} = \sin^{-1} x+C \\ \end{split}\]

Power Rule

Power Rule

For real number \(n\) where \(n\neq -1\)

\[\int x^n \; dx = \dfrac{1}{n+1}x^{n+1}+C\]

Algebraic Rules

\[ \displaystyle \int \big( f(x)+g(x)\big)\; dx =\int f(x) \; dx + \int g(x) \; dx \]

\[ \displaystyle \int \big( f(x)-g(x)\big)\; dx =\int f(x) \; dx - \int g(x) \; dx \]

\[ \displaystyle \int c\cdot f(x)\; dx =c \cdot \int f(x) \; dx \]