See the second bullet point: Trigonometric substitution - Wikipedia, the free encyclopedia
Since your integrand has 1^2 + x^2 in it, tangent is the appropriate trig substitution.
integrate[1/(x^2(x^2 +1)^0.5)] - Wolfram|Alpha
I don't entirely understand the first step. Could someone elaborate? I understand the basic idea of integration by sunstitution.
See the second bullet point: Trigonometric substitution - Wikipedia, the free encyclopedia
Since your integrand has 1^2 + x^2 in it, tangent is the appropriate trig substitution.