Antiderivative

A function $F$ is an antiderivative $f$ on an interval $l$ if $F^{\prime }(x)=f(x)$ for all $x$ in $l$ For example, position is the antiderivative of velocity.

Infinite Antiderivatives

If you want to find the antiderivative of a given $F$, you can create infinite $f$ functions.

Example

$v(t)=d^{\prime }(t)$ $v(t)=t^2$ $d(t)$ could be $\frac{1}{3}{t}^3+0,\frac{1}{3}{t}^3+1,\frac{1}{3}{t}^3+2,\dots$

Process

1. Increase power by 1 for each term
2. New coefficients are old coefficients divided by new power
3. Add constant $c$

Particular Antiderivative Table

$f(x)$	General $F(x)$
$1$	$x$
$x^n$	$(\frac{1}{n+1})x^{n+1}$
$\frac{1}{1+x^2}$	${\tan }^{-1}(x)$

Anti derivative Proofs

• Proving f’(0) has f as constant function
  • Proving Antiderivative is g(x) + c