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Antiderivatives

Engineering Context

Engineers are often interested in moving objects. Derivatives and antiderivatives enable engineers to find the position, velocity, and acceleration of these objects.

MAE: When a mechanical engineer is designing a shock absorber or a spring, antiderivatives can help them to calculate displacement. This makes it possible for mechanical engineers to design the most efficient spring possible.
ECE: Antiderivatives are what make our estimated time of arrival feature in our phones possible. Computers engineers would use antiderivatives to program a phone to calculate how long it will take to get from one location to the next when given a velocity.
BE: Antiderivatives can help biological engineers calculate how the concentration of a medicine in the blood stream changes over time. This would help them to develop pharmaceuticals that are both effective and safe.
CEE: Transportation engineers use antiderivatives to help them determine things like how long it takes to get from point A to point B at a given speed limit. This would help develop speed limits but also help them be able to study traffic trends if the actual travel time is different than their estimated travel time.

To learn more about the engineering applications of antiderivatives and related concepts, see the sections on derivatives and integration.

The Essentials

The antiderivative is the opposite or inverse of the derivative. In order to find the antiderivative, we need to do the opposite of what we did to find the derivative. For example, if we used the power rule to find that the derivative of x² is 2x, we need to essentially undo the power rule to find the antiderivative of 2x.

More generally, if $F (x)$ is the antiderivative of $f (x)$ , then

\int f (x) d x = F (x) + C

This + C is important because it represents a constant that may have been in the original equation before we took the derivative. Taking the derivative would have caused it to disappear. When we take an antiderivative it is important to include + C to compensate for any lost constant value.

Below is a table of commonly encountered types of functions and their antiderivatives:

Function	Antiderivative
$constant$	$x$
$x^{n}$	$\frac{x^{n + 1}}{n + 1}$
$\frac{1}{x}$	$\ln (x)$
$e^{x}$	$e^{x}$
$\cos (x)$	$\sin (x)$
$\sin (x)$	$- \cos (x)$

A Deeper Dive

After learning about finding antiderivatives, you may start to see similarties between finding antiderivatives and integrating. While they are quite similar, these terms do not have the exact same meaning. Anti-differentiation is simply the process of finding the antiderivative of a function. Integration is the process of finding the area under a curve. While some people are adamant about this technical difference, there are many who use these terms interchangeably.

In order to use integration to find the area under the curve we need to use a definite integral. Indefinite integration is integration over no specific bounds. This is comparable to antidifferentiation. When we take an indefinite integral we are given the antiderivative. Like mentioned above, it is important to include a + C on the end of the antiderivative to compensate for any constant value that would have been lost in taking the derivative.

Practice

Find the antiderivative of the following function.

h^{'} (x) = 3 x^{2} + 4 x + 6

Solution:

Using the table above, we know that the antiderivative of $x^{n}$ is equal to $\frac{x^{n + 1}}{n + 1}$ .

h (x) = \frac{3 x^{(2 + 1)}}{(2 + 1)} + \frac{4 x^{(1 + 1)}}{(1 + 1)} + 6 x