# Evaluating the integral $\int\frac{\mathrm dx}{\sqrt{|x|+a}}$

How to compute the following integral $$\int\frac{\mathrm dx}{\sqrt{|x|+a}}?$$

with $$a>0$$. I've been searching for hours in my books to find a pattern, but unfortunately with no success.

• Try the cases $x>0$ and $x<0$ separately, then see if you spot a relation afterwards? – John Doe Apr 10 at 14:28
• I'm not sure it makes sense to calculate the integral in the absence of some limits – Andrei Apr 10 at 14:28
• Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see Guidelines for good use of $\LaTeX$ in question titles. – Brian Apr 10 at 14:30

$$\int \frac{1}{\sqrt{\pm x+a}}$$
Where the sign depends on the sign is equal to the sign of $$x$$:
$$\int \frac{1}{\sqrt{\pm x+a}}=\int (\pm x+a)^{-\frac{1}{2}}=\pm 2(\pm x+a)^{\frac{1}{2}}+c$$
So the answer is: $$sgn(x) 2(|x|+a)^{\frac{1}{2}}+c$$