Calculus · Quick Reference

Integration Cheat Sheet

Integral Calculus · Lesson 9/20

In one line: integration is the reverse of differentiation — instead of finding a rate of change, it finds the original function (or accumulated total) from a rate.

Key Ideas

1Antiderivative. F(x) is an antiderivative of f(x) if F'(x) = f(x). Integration finds antiderivatives.

2Indefinite Integral Notation. Written as Integral of f(x) dx = F(x) + C, where C is an arbitrary constant since derivatives of constants are zero.

3The Power Rule for Integration. Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1.

4Constant Multiple and Sum Rules. Constants factor out of integrals, and integrals of sums split into sums of integrals — mirroring the rules for derivatives.

5Why '+C' Matters. Since many functions share the same derivative (they differ by a constant), every indefinite integral needs the '+C' to represent the whole family of antiderivatives.

Worked Examples

Find the integral of x^3 dx.

x^4/4 + C

Find the integral of (4x^2 - 3x + 5) dx.

(4x^3)/3 - (3x^2)/2 + 5x + C

Verify that F(x) = x^2 + 7 is an antiderivative of f(x) = 2x.

Confirmed: F(x) is a valid antiderivative

Formulas

Integral of x^n dx = x^(n+1)/(n+1) + C (n does not equal -1)

Integral of k dx = kx + C

Integral of [f(x)+g(x)] dx = Integral of f(x)dx + Integral of g(x)dx

Practice Yourself

Find the integral of x^5 dx.

x^6/6 + C

Find the integral of 7 dx.

7x + C

Find the integral of (2x + 3) dx.

x^2 + 3x + C