Calculating integral

Question

I need some help to solve this integral:

$$\int(x^2+y^2)^{-\frac32} \mathrm dx$$

Thank you.

Answer 1

Hint: try the substitution $x=y\tan \theta$.

Answer 2

Put $x = y \tan t$ and $dx = y \sec^2 t\ dt$, then $$ \begin{align} \int\frac{dx}{\sqrt{(x^2+y^2)^3}}&=\int\frac{y \sec^2 t\ dt}{\sqrt{(y^2\tan^2 t+y^2)^3}}\\ &=\int\frac{y \sec^2 t\ dt}{y^3\sec^3 t}\\ &=\frac{1}{y^2}\int\cos t\ dt\\ &=\frac{\sin t}{y^2}+C\\ &=\frac{x}{y^2\sqrt{x^2+y^2}}+C. \end{align} $$