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 e^{2x}: The derivative of e^{2x} x is 2e^{2x} 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.