Antiderivatives (the reverse of differentiation)

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

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

Examples

1. 01Which expression gives every function with this derivative?$\displaystyle 4x$AnswerHide

   Answer$\displaystyle 2x^2+C$
2. 02A function with this derivative can be written $ax^2+C$. What is $a$?$\displaystyle 10x$AnswerHide

   Answer$\displaystyle 5$
3. 03For a function with this derivative, what is □?$\displaystyle 6x^{2} \;\to\; \square x^{3}+C$AnswerHide

   Answer$\displaystyle 2$
4. 04Integrate$\displaystyle \int x^{4} \, dx$AnswerHide

   Answer$\displaystyle \frac{1}{5}x^{5} + C$
5. 05A function with this derivative can be written $ax^2+C$. What is $a$?$\displaystyle 2x$AnswerHide

   Answer$\displaystyle 1$
6. 06Integrate$\displaystyle \int x^{2} \, dx$AnswerHide

   Answer$\displaystyle \frac{1}{3}x^{3} + C$

FAQ

Q1What happens if I forget the +C?Show answerHide

A

The indefinite integral is incomplete. There are infinitely many functions whose derivative is $f(x)$, 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.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05