Integration of root x

Integration of √x dx. We have learned how to find the derivative of the square root of x on our page derivative of root x. In this article, we will discuss how to find the integration of the square root of x and solve a few related problems.

Integration of root x

What is the integration of root x? Let us find out.

Question: Evaluate $\int \sqrt{x} dx$

Solution:

Step 1: Write root x by the rule of indices.
\[\sqrt{x}=x^{1/2}\]
So we have

$\int \sqrt{x} dx=\int x^{1/2} dx$ $\cdots (i)$

Step 2: Use the following power rule formula of integration:
\[\int x^n dx =\dfrac{x^{n+1}}{n+1}+c\]
where $n$ is any real number. Applying this formula to (i), we obtain that

$\int \sqrt{x} dx =\int x^{1/2} dx$ $=\frac{x^{1/2+1}}{1/2+1}+c$

Step 3: Simplifying the above expression, we get

$\int \sqrt{x} dx =\dfrac{x^{1/2+1}}{1/2+1}+c$

$=\dfrac{x^{3/2}}{3/2}+c$

$=\dfrac{2}{3}x^{3/2}+c$

So the integration of the square root of x is

\[\int \sqrt{x} dx =\dfrac{2}{3}x^{3/2}+c\]
where $c$ is an integration constant.

Video Solution of the Integration of Root x:

 

Also Read: 

Integration of mod x: The integration of  |x| is -x|x|/2 +c

Derivative of root x: The derivative of √x is 1/2√x

Derivative of e2x: The derivative of e2x x is 2e2x

Derivative of cube root of x: The derivative of the cube root of x is 1/(3x^{2/3})

Integration of 1/root x

Question: Evaluate $\int \frac{1}{\sqrt{x}} dx$

Solution:

Using the same to above, we can find the integration of 1/root(x) dx. We have:

$\int \dfrac{1}{\sqrt{x}} dx$

$=\int x^{-1/2} dx$ (by the power rule of exponents)

$=\dfrac{x^{-1/2+1}}{-1/2+1}+c$

$[\because \int x^n dx =\dfrac{x^{n+1}}{n+1}]$

$=\dfrac{x^{1/2}}{1/2}+c$

$=2x^{1/2}+c$

$=2\sqrt{x}+c$

So the integration of root x is

\[\int \sqrt{x} dx =2\sqrt{x}+c\]
where $c$ is an integration constant.

Video Solution of the Integration of 1/root x:

Integration of root x+(1/root x)

Question: Find the integration of $\sqrt{x}+\frac{1}{\sqrt{x}}$

Solution:

$\int \big(\sqrt{x}+\dfrac{1}{\sqrt{x}} \big) dx$

$=\int \sqrt{x} dx + \int \frac{1}{\sqrt{x}} dx$

$=\dfrac{2}{3}x^{3/2}+2\sqrt{x}+c$

where $c$ is an integration constant.

Integration of x root x

Question: Find the integration of $x\sqrt{x}$

Solution:

$\int x\sqrt{x} dx$

$=\int x \cdot x^{1/2} dx$

$=\int x^{1+1/2} dx$

$=\int x^{3/2} dx$

$=\dfrac{x^{3/2+1}}{3/2+1}+c$

$[\because \int x^n dx =\dfrac{x^{n+1}}{n+1}]$

$=\dfrac{x^{5/2}}{5/2}+c$

$=\dfrac{2}{5}x^{5/2}+c$

where $c$ is an integration constant.

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