Steps to solve $\int \sqrt{\frac{11}{x}}\,\mathrm{d}x$?

Question

What are the steps required to solve the following?

$\int \sqrt{\frac{11}{x}}\,\mathrm{d}x$

I'm not looking for anyone to do my homework. I usually have no problem figuring these things out -- using Wolfram Alpha step-by-step if absolutely necessary -- but for some reason this seemingly simple problem has me stumped.

Below are the Wolfram Alpha step-by-step instructions for doing this. I get lost on the part where they do the u substitutions.

Any help is greatly appreciated.

Answer 1

There's no need for a substitution, and that only complicates matters and obscures what's important. Write

$$\sqrt{\frac{11}{x}} = \frac{\sqrt{11}}{\sqrt x} = \sqrt{11} x^{-1/2}$$

Now integrate:

\begin{align*} \int \sqrt{11} x^{-1/2} dx &= \sqrt{11} \int x^{-1/2} dx \\ &= \sqrt{11} \frac{x^{-1/2 + 1}}{-1/2 + 1} + C \\ &= \sqrt{11} \frac{x^{1/2}}{1/2} + C \\ &= 2 \sqrt{11} x^{1/2} + C \\ &= 2 \sqrt{11 x} + C \end{align*}

Answer 2

Suggestion $\frac{1}{\sqrt{x}} = x^{-1/2}$.

Answer 3

Hint:

Remove the constant. Then note that $\sqrt\frac{1}{x} = \frac{1}{\sqrt{x}} = x^{\frac{-1}{2}}$.

You should be able to go from here simply applying the Fundamental Theorem of Calculus. The derivative of what will give you $x^{\frac{-1}{2}}$?