Question

What is the antiderivative of $2x$?

Answer 1

An antiderivative is nothing but the derivative of the function $F$ whose derivative is equal to the original function i.e. $F’ \left( x \right)=f\left( x \right)$, then $F\left( x \right)$ is an antiderivative of $f\left( x \right)$. The general antiderivative is $F\left( x \right)+C$ of $f\left( x \right)$.

Complete step-by-step solution:

Now let us find out the antiderivative of $2x$.
According to the rule of finding out the antiderivative, there exists a number of antiderivatives of $2x$.
The general antiderivative of $2x$ is $~{{x}^{2}}+C$ (according to the power rule).
So every function that can be expressed in the form of $~{{x}^{2}}$ and a constant is a derivative for $2x$.
To clear it, we can consider certain examples. For example if we take functions as - ${{x}^{2}},~{{x}^{2}}+9,~{{x}^{2}}-69, ~~{{x}^{2}}+17\pi 8-\surd 21$ etc then these are the functions that can be expressed in the form of antiderivative of $2x$.
But the most common antiderivative would be of ${{x}^{2}}+C$.
We can notice that the power of the variable in the antiderivative is the same as the constant of the function whose antiderivative is to be found out.
$\therefore$ The antiderivative of $2x$ is ${{x}^{2}}+C$.

Note: Generally, the antiderivative is expressed as the derivative along with the arbitrary constant $C$. Any of the functions can be considered for finding out the antiderivative but the most convenient one would be the general antiderivative.
This can also be solved mathematically in the following way-
The antiderivative of power rule is $\int {{x}^{n}}\text{d}x=\dfrac{1}{n+1}{{x}^{n+1}}+C$
Just as $\dfrac{\text{d}}{\text{d}x}(af(x))=a\dfrac{\text{d}}{\text{d}x}f(x)$
$\therefore$ We can perform the same to the function we have and we will get

& \int 2xdx \\\ & =2\int xdx \\\ & =2\int {{x}^{1}}dx \\\ & =2\dfrac{1}{1+1}{{x}^{1+1}}+C \\\ & =2\dfrac{1}{2}{{x}^{2}}+C \\\ & ={{x}^{2}}+C \\\ \end{aligned}$$ Hence the antiderivative is obtained mathematically.