Calculate $\int{\frac{x^\alpha}{(x^2+a^2)^{\frac{3}{2}}}} \ \mathrm{d}x$ for $\alpha = 1,2,4,6$? What is the most straightforward way to calculate the following indefinite integral? $$\int{\frac{x^\alpha}{(x^2+a^2)^{\frac{3}{2}}}}\ \mathrm{d}x,$$
for $\alpha = 1, 2, 4,5,6$?

HINT: use substitution $x=a\tan \theta$, 
$$\int \frac{x^{\alpha}}{(x^2+a^2)^{3/2}}dx=\int \frac{a^{\alpha}\tan^{\alpha}\theta}{\sec^3\theta}\sec^2\theta\ d\theta=a^{\alpha}\int \frac{\tan^{\alpha}\theta}{\sec\theta} d\theta $$
