Integrate $\int \frac{dx}{(x^2+c)^\frac{3}{2}}$

Question

Using Mathematica, I found a simple result

$$ \int \frac{dx}{(x^2+c)^\frac{3}{2}} = \frac{x}{c\sqrt{x^2+c}} + const$$

where $c$ is a constant.

But I am unable to get this result by hand - I don't know what method to use. Any ideas?

Answer 1

HINT:

If $c>0$ set $x=\sqrt c\tan\theta$

If $c<0,c=-d$(say) set $x=\sqrt d\sec\theta$

See : Trigonometric substitutions

Answer 2

Hint

Use substitution $x=\sqrt c \sinh(y)$ so $dx=\sqrt c \cosh(y)$ and $$I=\int \frac{dx}{(x^2+c)^\frac{3}{2}} =\frac 1c \int \frac{dy}{\cosh^2(y)}=\tanh(y)$$ Now, go back to $x$.

I am sure that you can take from here.