Solve $\int \frac{dz}{(A^2+z^2)\sqrt{2A^2+z^2}}$

Question

Let A be a constant and z a variable.

Compute the integral:

$\int \frac{dz}{(A^2+z^2)\sqrt{2A^2+z^2}}$

Note: I've tried the most common trigonometric substitutions (like z = Atan($\theta$)), but had no success.

Answer 1

Let $z=\sqrt{2}A\tan \theta$

The integral becomes (I will just skip the trivial steps) $$\int \frac{\cos\theta d\theta}{A^2(1+\sin^2\theta)} = \frac{1}{A^2}\int \frac{ d\sin\theta}{1+\sin^2\theta} = \frac{\arctan(\sin\theta)}{A^2}$$

I think this is clear enough and you can figure out the rest.

Answer 2

you Can use the Substitution $$\sqrt{2A^2+z^2}=t+z$$ then you will get $$z=\frac{2A^2-t^2}{2t}$$ and $$dz=-\frac{2 A^2+t^2}{2 t^2}dt$$