3.9: Antiderivatives

If you had a function $s = f(t)$ representing distance traveled, the derivative $f'(t)$ gives you velocity. What if you knew $f'(t)$ and wanted to find $f(t)$?

In this situation, $f(t)$ is called an antiderivative of $f'(t)$. The definition follows.

A function $F$ is called an antiderivative of $f$ on an interval $I$ if $F'(x) = f(x)$ for all $x \in I$.
Find three different antiderivatives for $f(x) = x^2$.

Recall this corollary from the Mean Value Theorem:

Corollary: If $f'(x) = g'(x)$ for all $ x \in (a, b)$, then $f - g$ is constant on $(a, b)$, that is, \[f(x) = g(x) + c\] where $c$ is a constant.

This tells us all antiderivatives of $f(x)$ must only differ by a constant:

If $F$ is an antiderivative of $f$ on an interval $I$, then the most general antiderivative of $f$ on $I$ is \[F(x) + C\] where $C$ is an arbitrary constant.
Find the most general antiderivative of the following functions:
  1. $f(x) = x^2$
  2. $f(x) = \sin x$
  3. $f(x) = x^n, \ n \geq 0$
  4. $f(x) = x^2$
Find all functions $g(x)$ where \[g'(x) = 4\sin x + \dfrac{2x^5 - \sqrt{x}}{x}\]
Sometimes you will need to find the constant $C$ in $f(x)$. In order to find $C$, you will need an additional piece of information to solve for $C$. Such a situation may come up later in your mathematical career and problems of this form are called differential equations (these are equations involving the derivatives of a function).
Find $f$ if $f'(x) = x\sqrt{x}$ and $f(1) = 2$.