Integrals: the basics

Antiderivatives (the reverse of differentiation)

An antiderivative of f(x)f(x) is a function F(x)F(x) whose derivative is f(x)f(x); together they form the indefinite integral ∫f(x) dx=F(x)+C\int f(x)\,dx=F(x)+C.

F′(x)=f(x)  ⟺  ∫f(x) dx=F(x)+C\begin{gathered} F'(x)=f(x) \\[4pt] \iff \int f(x)\,dx=F(x)+C \end{gathered}

Integration is the reverse of differentiation. Finding a function whose derivative is f(x)f(x) is the indefinite integral, and such a function is an antiderivative.

F′(x)=f(x)   ⟺   ∫f(x) dx=F(x)+CF'(x)=f(x)\ \iff\ \int f(x)\,dx=F(x)+C

Constants vanish under differentiation, so antiderivatives are free up to a constant CC. Always write +C+C in the answer.

Examples

  1. 01Which expression gives every function with this derivative?4x\displaystyle 4xAnswerHide
    Answer2x2+C\displaystyle 2x^2+C
  2. 02A function with this derivative can be written ax2+Cax^2+C. What is aa?10x\displaystyle 10xAnswerHide
    Answer5\displaystyle 5
  3. 03For a function with this derivative, what is □?6x2  →  □x3+C\displaystyle 6x^{2} \;\to\; \square x^{3}+CAnswerHide
    Answer2\displaystyle 2
  4. 04Integrate∫x4 dx\displaystyle \int x^{4} \, dxAnswerHide
    Answer15x5+C\displaystyle \frac{1}{5}x^{5} + C
  5. 05A function with this derivative can be written ax2+Cax^2+C. What is aa?2x\displaystyle 2xAnswerHide
    Answer1\displaystyle 1
  6. 06Integrate∫x2 dx\displaystyle \int x^{2} \, dxAnswerHide
    Answer13x3+C\displaystyle \frac{1}{3}x^{3} + C

FAQ

What happens if I forget the +C?
The indefinite integral is incomplete. There are infinitely many functions whose derivative is f(x)f(x), differing by a constant, and C represents all of them. In a definite integral C cancels in the subtraction, so it is not needed there.

Published 2026-09-03 · Updated 2026-10-14