Calculus · Quick Reference

Indefinite Integrals Cheat Sheet

Integral Calculus · Lesson 11/20

In one line: an indefinite integral represents an entire family of antiderivatives, differing only by a constant, rather than a single numeric value.

Key Ideas

1Family of Functions. Integral of f(x) dx = F(x) + C represents infinitely many functions, one for each possible value of C.

2Geometric View. All antiderivatives of the same function are vertical shifts of one another — they share identical slopes at every x.

3Finding a Particular Antiderivative. If given an initial condition (like F(0) = 5), substitute it into F(x) + C to solve for the specific value of C.

4Basic Integral Rules Recap. Power rule, constant multiple rule, and sum rule all apply the same way as with definite integrals, just without evaluating at bounds.

5Common Integral Forms. Integral of e^x dx = e^x + C. Integral of 1/x dx = ln|x| + C. Integral of sin(x) dx = -cos(x) + C. Integral of cos(x) dx = sin(x) + C.

Worked Examples

Find the indefinite integral of (6x^2 + 4) dx.

2x^3 + 4x + C

Find F(x) if F'(x) = 3x^2 and F(1) = 5.

F(x) = x^3 + 4

Find the integral of (cos(x) + e^x) dx.

sin(x) + e^x + C

Formulas

Integral of e^x dx = e^x + C

Integral of 1/x dx = ln|x| + C

Integral of sin(x) dx = -cos(x) + C

Integral of cos(x) dx = sin(x) + C

Practice Yourself

Find the integral of (10x + 2) dx.

5x^2 + 2x + C

Find F(x) if F'(x)=2x and F(0)=3.

F(x) = x^2 + 3

Find the integral of sin(x) dx.

-cos(x) + C