Integration by Reduction Formulae

[saladfinger16]

Can anyone help me out with a proof for the integral that's in the attached images, its driving me nuts

 

[snipez90]

You could try the trig substitution x = a*tan(u).

Or note that

$$\int\frac{dx}{(x^2 + a^2)^n} = \int\frac{dx}{(x^2 + a^2)^{n-1}} - \int\frac{x^2 + a^2 - 1}{(x^2 + a^2)^n}\,dx$$

and try to work out the last integral on the right.