Primitives of common functions

Table of common antiderivatives

Function f	Antiderivative F	Interval
f(x) = a (constant)	F(x) = a*x	R
f(x) = x^n (n an integer, n ≥ 1)	F(x) = x^(n+1)/(n+1)	R
f(x) = 1/x²	F(x) = -1/x	]0;+∞[ or ]-∞;0[
f(x) = 1/√x	F(x) = 2*√x	]0;+∞[
f(x) = 1/x	F(x) = ln(x)	]0;+∞[
f(x) = e^x	F(x) = e^x	R
f(x) = cos(x)	F(x) = sin(x)	R
f(x) = sin(x)	F(x) = -cos(x)	R

Linearity

If F is an antiderivative of f and G is an antiderivative of g, then:

• F + G is an antiderivative of f + g
• kF is an antiderivative of kf (where k is real)

Example: an antiderivative of 5x^2 + 3 is 5*(x^3/3) + 3x = (5/3)x^3 + 3x.

Detailed example

Find an antiderivative of f(x) = 4x³ - 2x + 7 on ℝ. We deal with each term: the antiderivative of 4x³ is x⁴, the antiderivative of -2x is -x², and the antiderivative of 7 is 7x. Therefore, F(x) = x⁴ - x² + 7x + C.

Common pitfall

Do not confuse the antiderivative of x^n with its derivative: we ADD 1 to the exponent and DIVIDE by this new exponent (the inverse of the differentiation rule, which multiplies and subtracts 1). Always check by differentiating F to find f.