Integral from Cambridge STEP Paper.

Show, by means of a suitable change of variable, or otherwise, that

$\displaystyle

\int_0^\infty f((x^2+1)^{1/2} +x) dx = \frac{1}{2}\int_1^\infty (1+t^{-2})f(t) dt

$

Hence, or otherwise, show that

$\displaystyle

\int_0^\infty ((x^2+1)^{1/2} +x)^{-3} dx = \frac{3}{8}

$

This is taken directly from STEP 1, 1998. Now the challenge isn't finding a solution, that's cake for many of you, the challenge is explaining it to me. A maths student in the first year of sixth-form (I think you call it high-school in America.) I've never seen integrals like these before, the most advanced form of integration I've used is integrations by parts.

Please explain EVERYTHING!