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.

  • 2
    $\begingroup$ Try the cases $x>0$ and $x<0$ separately, then see if you spot a relation afterwards? $\endgroup$ – John Doe Apr 10 '19 at 14:28
  • $\begingroup$ I'm not sure it makes sense to calculate the integral in the absence of some limits $\endgroup$ – Andrei Apr 10 '19 at 14:28
  • $\begingroup$ 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. $\endgroup$ – Brian Apr 10 '19 at 14:30

We can rewrite it as follows:

$$\int \frac{1}{\sqrt{\pm x+a}}$$

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: $$sgn(x) 2(|x|+a)^{\frac{1}{2}}+c$$


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