Calculus / Unit 7: Antiderivatives and the Integral
7.1 Antiderivatives
Notes
An antiderivative of $f$ is a function $F$ with $F' = f$.
- Antiderivatives are not unique: if $F$ is one, so is $F + C$ for any constant $C$.
- The family of all antiderivatives is the indefinite integral: $\displaystyle \int f(x)\,dx = F(x) + C$.
- Check an answer by differentiating it.
- Example: an antiderivative of $4x^3$ is $x^4$, since $(x^4)' = 4x^3$.
Practice
Which is an antiderivative of $4x^3$?
$(x^4)' = 4x^3$.
Which function is NOT an antiderivative of $2x$?
$(2x^2)' = 4x$. The others differ from $x^2$ only by a constant.
Why do indefinite integrals include $+ C$?
Constants vanish when differentiating, so all of $F + C$ are antiderivatives.
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