Integrating $\int \frac{u \,du}{(a^2+u^2)^{3/2}}$ How does one integrate $$\int \frac{u \,du}{(a^2+u^2)^{3/2}} ?$$
Looking at it, the substitution rule seems like method of choice. What is the strategy here for choosing a substitution?

$$\int\frac{u}{\left(a^2+u^2\right)^{\frac{3}{2}}}\space\text{d}u=$$

Substitute $s=a^2+u^2$ and $\text{d}s=2u\space\text{d}u$:

$$\frac{1}{2}\int\frac{1}{s^{\frac{3}{2}}}\space\text{d}s=\frac{1}{2}\int s^{-\frac{3}{2}}\space\text{d}s=\frac{1}{2}\cdot-\frac{2}{\sqrt{s}}+\text{C}=-\frac{1}{\sqrt{a^2+u^2}}+\text{C}$$
