# What is the antiderivative of 2x?

Sep 1, 2015

There are many antiderivatives of $2 x$. The (most) general antiderivative is ${x}^{2} + C$

#### Explanation:

An antiderivative of $2 x$ is a function whose derivative is $2 x$:

antiderivatives include:

${x}^{2}$, $\text{ } {x}^{2} + 7$, $\text{ } {x}^{2} + 19$, $\text{ } {x}^{2} - 11$, $\text{ } {x}^{2} + \frac{17 \pi}{8} - \sqrt{21}$

${x}^{2} + {\sin}^{2} x + {\cos}^{2} x$

Any (every) function that can be expressed in the form ${x}^{2} + \text{some constant}$ is an antiderivative.

The general antiderivative is expressed by choosing one of the antiderivatives and adding an "arbitrary constant" usually named $C$

It is convenient (but not required) to choose the first antiderivative on the list above and say:

The (most) general antiderivative of $2 x$ is ${x}^{2} + C$.

Strange but true
According to the definition of general antiderivative, we can also say "The (most) general antiderivative of $2 x$ is ${x}^{2} + {\sin}^{2} x + {\cos}^{2} x + 19 \sqrt{\pi} - \frac{\sqrt{37}}{4} + C$"

Important! -- check you textbook's definition (and your grader's sense of humor) before using this smart-alecky answer on an exam!