Question

What is the antiderivative of $\sqrt{x}$?

Answer 1

To do this question, you need to know the formula for finding the antiderivative of ${{x}^{n}}$. The anti derivative of ${{x}^{n}}$ is $\dfrac{{{x}^{n+1}}}{n+1}$. Therefore, by using this formula, you can find the antiderivative of $\sqrt{x}$. Here you should take n as $\dfrac{1}{2}$. Therefore, you should now substitute the value of n and get the final answer. Also add the constant which is important.

Complete step by step solution:
Here is the step wise solution.
First we need to write down the formula for the antiderivative of ${{x}^{n}}$.
The anti derivative of ${{x}^{n}}$ is $\dfrac{{{x}^{n+1}}}{n+1}$.

In the question, we are asked to find the antiderivative for the ${{x}^{n}}$. But we know that ${{x}^{n}}$ can also be written as ${{x}^{\dfrac{1}{2}}}$. So, now we can use the formula of antiderivative of ${{x}^{n}}$. Hence, we get n as $\dfrac{1}{2}$. Therefore we get the anti derivative of ${{x}^{n}}$ as:
$\Rightarrow \int{\sqrt{x} dx=}\dfrac{{{x}^{\dfrac{1}{2}+1}}}{\dfrac{1}{2}+1}+c$.
$\Rightarrow \int{\sqrt{x} dx=}\dfrac{{{x}^{\dfrac{3}{2}}}}{\dfrac{3}{2}}+c$.
$\Rightarrow \int{\sqrt{x} dx=}\dfrac{2}{3}x\sqrt{x}+c$.

Therefore, we get the final answer for the questions, how do you find anti derivative of ${{x}^{n}}$ as $\dfrac{2}{3}x\sqrt{x}+c$.

Note: Here, you should know what anti derivative is, to solve this question. Anti derivative is also known as the integration, which is the opposite of differentiation. To check your answer, you can differentiate your answer. If you get the answer the same as the question, then your answer is correct, otherwise you have to see where you went wrong. It is important that you write the constant of integration as this is an indefinite integration.