Question

How do you integrate `int dx/(ax^2+x^2)^(3/2)` using trig substitutions?

Answer 1

`(-1)/(2x^2(a+1)^(3/2))+C`

Explanation:

Using a trig substitution would not work here. However, simplifying this reveals that there is a simpler, yet sneakier, solution.

`intdx/(ax^2+x^2)^(3/2)=intdx/(x^2(a+1))^(3/2)=intdx/((x^2)^(3/2)(a+1)^(3/2))`

Note that `(a+1)^(3/2)` is a constant:

`=1/(a+1)^(3/2)intdx/x^3=1/(a+1)^(3/2)intx^-3dx`

`=1/(a+1)^(3/2)((x^-2)/(-2))=(-1)/(2x^2(a+1)^(3/2))+C`