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