Antiderivative Formulas

We will use the notation ∫ f(x)dx to denote the antiderivative of f(x). In the following formulas, f(x) and g(x) are functions and c is a constant:

∫ f + gdx = ∫ f + ∫ g
∫ f − gdx = ∫ f − ∫ g
∫ cfdx = c ∫ fdx
∫ f′(g(x))g′(x) = f(g(x)) + C

Formulas for common functions:

∫ x^{a} dx = \frac{x^{a+1}}{a+1} + C if a ≠ −1  ∫ x^{−1} dx = ln x + C
∫ a^{x} dx = \frac{a^{x}}{ln a} + C  ∫ ln x dx = x ln x − x + C
∫ sin x dx = − cos x + C  ∫ cos x dx = sin x + C
∫ tan x dx = − ln | cos x| + C  ∫ cot x dx = ln | sin x| + C
∫ sec x dx = ln | sec x + tan x| + C ∫ csc x dx = ln | csc x − cot x| + C
∫ sec^{2} x dx = tan x + C  ∫ csc^{2} x dx = − cot x + C
∫ sec x csc x dx = sec x + C  ∫ csc x cot x dx = − csc x + C
∫ \frac{1}{\sqrt{1−x^{2}}} dx = arcsin x + C  ∫ \frac{1}{x^{2}+1} dx = arctan x + C

∫ \frac{1}{x\sqrt{x^{2}−1}} dx = arcsec x + C

∫ sinh x dx = cosh x + C  ∫ cosh x dx = sinh x + C