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How do you find the antiderivative of $f(x)=3{{x}^{2}}+2$?

Last updated date: 11th Sep 2024
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Hint: We are given a function $f(x)=3{{x}^{2}}+2$ we have asked to find the derivative of f(x), we will learn what are anti derivative, we will learn how derivative and integration if connected to each other we will use the
$\int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{x+1}}$
we will also use that
$\int{k{{x}^{x}}dx=k\int{{{x}^{x}}dx}}$
We will need that $1={{x}^{0}}$
Using this information, we will find integral of $f(x)=3{{x}^{2}}+2$

Complete step-by-step solution:
We are given a function as $f(x)=3{{x}^{2}}+2$ we are asked to find the antiderivative, before we move forward, we will understand what are anti derivative, we will understand what we are asked to find.
Anti-derivative as the name suggests the anti (opposite) of derivative, there is a method called integral which is also known as anti-derivate. In this we will add the small pieces together to find the bigger term.
So, we have $f(x)=3{{x}^{2}}+2$
We have to find the integral, to do so we will use
$\int{{{x}^{n}}dx=\dfrac{{{x}^{x+1}}}{x+1}}$
$\int{=3{{x}^{2}}+2}=\int{(3{{x}^{2}})dx+\int{2dx}}$
As $\int{(a+b)dx=\int{adx+\int{bdx}}}$
$=3\int{{{x}^{2}}dx+2\int{dx}}$
using property $\int{k{{x}^{x}}dx=k\int{{{x}^{x}}dx}}$
Now we integrate $\int{{{x}^{2}}dx}\,\,\text{and}\,\,\int{idx}$
\begin{align} & =3\left( \dfrac{{{x}^{3}}}{3} \right)+2\left( \dfrac{x}{1} \right)+c \\ & \int{{{x}^{2}}=\dfrac{{{n}^{2+1}}}{2+1}\,\,\,\operatorname{and}\,\,\int{idx=\int{x0dx=\dfrac{{{x}^{0+1}}}{0+1}}}} \\ \end{align}
Simplifying we get
${{x}^{3}}+2x+c$
when c is constant
So, anti-derivative of $f(x)=3{{x}^{2}}+2$ is ${{x}^{3}}+2x+c$

Note: Remember that we can check our solution by first finding the derivative of ${{x}^{3}}+2x+c$ also remember that, if the term involving the substitution, then it is necessary to change the substitution back to the basic integral identity (without limit) and if it is definite (with limit) then we need to change limit as well.