21. Antiderivatives, Areas and the FTC

We have learned how to take derivatives. We now learn how to do this backwards.

a1. Definition of Antiderivatives

An antiderivative of a function \(f(x)\) is any function \(F(x)\) whose derivative is \(f(x)\): \[ \dfrac{dF}{dx}=f(x) \]

We tend to use lower case letters for the original function and the corresponding upper case letter for the antiderivative.

Find an antiderivative of \(f(x)=x^3\).

We know that the derivative of \(x^4\) is \(4x^3\). This is \(4\) times too much. So we take \(\dfrac{1}{4}\) of it. So an antiderivative of \(f(x)=x^3\) is \(F(x)=\dfrac{1}{4}x^4\).

We check this by differentiating: \[ \dfrac{d}{dx}\left[\dfrac{1}{4}x^4\right]=\dfrac{1}{4}(4x^3)=x^3 \]