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Can someone show me how to get the following integral please. I'm not sure how to approach this problem

$$ \int \frac{dx}{2 + \sqrt{x}} $$

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@barrycipra I did! I also deleted my comment as the answer was already here before I made my comment :). Thanks – Chinny84 Feb 24 '14 at 14:00
What are (or were) your thoughts so far? We could help you a lot more in future if you show us exactly what you've tried and where you're stuck :) – Shaun Feb 24 '14 at 14:28
up vote 9 down vote accepted

Try making the following substitution:

$$\sqrt x = u \implies x = u^2 \text{ and } dx = 2u\,du$$

That gives you the integral $$\begin{align} \int \frac{dx}{2 + \sqrt x} & = \int \frac{2u\;du}{2+u}\tag{substitution $u = \sqrt x$} \\ \\ & = \int \left(2 - \frac{4}{2+u}\right)\,du\tag{polynomial division}\end{align}$$

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Even better would be $u=2+\sqrt{x}$, i.e., $x=(u-2)^2$ for $u\ge 2$. – LutzL Feb 24 '14 at 13:57

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