Integrals · Definition

Antiderivative

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Before you start you need to know

An antiderivative reverses differentiation: given \(f\), we look for \(F\) with \(F'=f\). Here you will learn to check it by differentiating and why a constant \(+C\) always appears.

Definition

An antiderivative of \(f\) is a function \(F\) such that \(F^{\prime}=f\). If \(F\) is an antiderivative of \(f\), all the others on an interval differ from \(F\) by a constant.

\[F'(x)=f(x)\]

An antiderivative of \(f\) is a function \(F\) whose derivative is \(f\). It is "differentiating backwards": given velocity, recover position.

It is not unique: if \(F\) is one, so is \(F+C\) for any constant \(C\), since the derivative of a constant is 0. That is why integrating always brings a "+C".

Every differentiation formula reads backwards as an antidifferentiation formula, though not every antiderivative can be written with elementary functions.

An antiderivative is also called a primitive.

Think about it

Why do antiderivatives include a constant C?

Because functions differing by a constant have the same derivative.

Exercises 10

  1. If \(F\) and \(G\) are antiderivatives of the same function on an interval, then...
  2. Which is an antiderivative of \(f(x)=\cos {x}\)?
  3. Which is an antiderivative of \(f(x)=\frac{1}{x}\)?
  4. Which is an antiderivative of \(f(x)=\sec^{2} {x}\)?
  5. Which is an antiderivative of \(f(x)=x^{2}\)?
  6. Which is an antiderivative of \(f(x)=e^{x}\)?
  7. Which is an antiderivative of \(f(x)=\sin {\left(2x \right)}\)?
  8. Let \(f(x)=0\) on \(D=(-1,0)\cup(0,1)\). Which statement is true?
  9. Which is an antiderivative of \(f(x)=\frac{1}{x^{2}}\)?
  10. Which is an antiderivative of \(f(x)=x e^{x^{2}}\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Tom M. Apostol, Calculus, Vol. 1, 2.ª edición, John Wiley & Sons, 1967.