The past few chapters have discussed determining the derivative of a given function. Assume for a moment that we are given the derivative of a polynomial function. Is it possible to determine the original polynomial function? The answer is yes and this is accomplished by applying the Anti-Differentiation Power Rule, or essentially, The Power Rule in reverse. Recall The Power Rule:

Simply put, we multiplied the coefficient by the exponent, then subtracted one from the exponent. If we want to reverse this process, then it would make sense to add one to the exponent first, then divide the coefficient by the new exponent.

Given a polynomial of the form: $f′(x)= ax^n$ then an antiderivative is:

$f(x)=\dfrac{a}{n+1}x^{n+1}$

This can be seen in the following example term by term.

Example 1: Let $f'(x)=4x^2+6x−2$. Determine $f(x)$.

Explanation: $f(x)=\dfrac{4x^3}{3} + \dfrac{6x^2}{2} − \dfrac{2x}{1}=\dfrac{4}{3}x^3+3x^2−2x$. It is worth checking this answer by using The Power Rule. However, this is not the complete picture. In reality, there are infinitely many functions that have a derivative of $f'(x)=4x^2+6x−2$. They are, to name a few:

$\dfrac{4}{3}x^3+3x^2−2x+1$,
or $\dfrac{4}{3}x^3+3x^2−2x+2$,
or $\dfrac{4}{3}x^3+3x^2−2x+3$
or $\dfrac{4}{3}x^3+3x^2−2x+4,…$
and so on.

Remember, the derivative of any constant is zero. Thus, the answer to the previous question is $f(x)=\dfrac{4}{3}x^3+3x^2−2x+c$ where $c \in \mathbb R$.

Example 2: Let $f'(x)=4x^3+3x^2−1$. Determine a function $f(x)$ that goes through the point $(1,8)$.

Explanation: We should use the anti-derivative power rule first. However, this function has the initial condition that it travels through the point $(1,8)$.

Due to this condition, we are looking for one specific function. In other words, the constant $c$ can be determined.

$f(x)=x^4+x^3−x+c$
$\Rightarrow 8=14+13−1+c$
$\Rightarrow 8=1+c$
$\Rightarrow c=7$.

Thus, $f(x)=x^4+x^3−x+7$.

Since the derivative of other common functions was discussed in a previous chapter, the anti-derivative of other common functions are shown below with 𝑎 and 𝑏 as constants.

$f'(x)$ $f(x)$
$a\sin(bx)$ $-\dfrac{a}{b}\cos(bx)+c$
$a\cos(bx)$ $\dfrac{a}{b}\sin(bx) + c$
$a\mathrm e^{bx}$ $\dfrac{a}{b}\mathrm e^{bx}+c$
$\dfrac{a}{x}$ $a\ln(bx) + c$