Calculus I

Indefinite Integrals

In view of the fundamental theorem of calculus the indefinite integral $\displaystyle\int f(x) dx = F(x) + C$ is an antiderivative in the sense that $\displaystyle\frac{d}{dx}[F(x) + C] = f(x)$

Functions Indefinite integrals (antiderivatives)

Power and logarithmic functions $\displaystyle\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ( $n \neq -1$ )
$\displaystyle\int \frac{1}{x} dx = \ln \vert x\vert + C$

Exponential functions $\displaystyle\int e^x dx = e^x + C$

Trigonometric functions $\displaystyle\int \cos x dx = \sin x + C$
$\displaystyle\int \sin x dx = -\cos x + C$
$\displaystyle\int \sec^2 x dx = \tan x + C$
$\displaystyle\int \csc^2 x dx = -\cot x + C$
$\displaystyle\int \sec x \tan x dx = \sec x + C$
$\displaystyle\int \csc x \cot x dx = -\csc x + C$

Hyperbolic functions $\displaystyle\int \cosh x dx = \sinh x + C$
$\displaystyle\int \sinh x dx = \cosh x + C$
$\displaystyle\int \frac{1}{\cosh^2 x} dx = \tanh x + C$
$\displaystyle\int \frac{1}{\sinh^2 x} dx = -\coth x + C$

Inverse trigonometric functions $\displaystyle\int \frac{1}{\sqrt{1 - x^2}} dx = \sin^{-1} x + C$
$\displaystyle\int \frac{1}{1 + x^2} dx = \tan^{-1} x + C$

Inverse hyperbolic functions $\displaystyle\int \frac{1}{\sqrt{1 + x^2}} dx = \sinh^{-1} x + C$
$\displaystyle\int \frac{1}{\sqrt{x^2 - 1}} dx = \cosh^{-1} x + C$
$\displaystyle\int \frac{1}{1 - x^2} dx = \tanh^{-1} x + C$

Functions	Indefinite integrals (antiderivatives)
Power and logarithmic functions	$\displaystyle\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ( $n \neq -1$ ) $\displaystyle\int \frac{1}{x} dx = \ln \vert x\vert + C$
Exponential functions	$\displaystyle\int e^x dx = e^x + C$
Trigonometric functions	$\displaystyle\int \cos x dx = \sin x + C$ $\displaystyle\int \sin x dx = -\cos x + C$ $\displaystyle\int \sec^2 x dx = \tan x + C$ $\displaystyle\int \csc^2 x dx = -\cot x + C$ $\displaystyle\int \sec x \tan x dx = \sec x + C$ $\displaystyle\int \csc x \cot x dx = -\csc x + C$
Hyperbolic functions	$\displaystyle\int \cosh x dx = \sinh x + C$ $\displaystyle\int \sinh x dx = \cosh x + C$ $\displaystyle\int \frac{1}{\cosh^2 x} dx = \tanh x + C$ $\displaystyle\int \frac{1}{\sinh^2 x} dx = -\coth x + C$
Inverse trigonometric functions	$\displaystyle\int \frac{1}{\sqrt{1 - x^2}} dx = \sin^{-1} x + C$ $\displaystyle\int \frac{1}{1 + x^2} dx = \tan^{-1} x + C$
Inverse hyperbolic functions	$\displaystyle\int \frac{1}{\sqrt{1 + x^2}} dx = \sinh^{-1} x + C$ $\displaystyle\int \frac{1}{\sqrt{x^2 - 1}} dx = \cosh^{-1} x + C$ $\displaystyle\int \frac{1}{1 - x^2} dx = \tanh^{-1} x + C$