Help! Derive exact solution to the definite integral

Integral from 0 -> (sqrt 2) / 2

( x^2 / sqrt(1-x^2) ) dx

∫x²/√(1 - x²) dx from 0 to √2/2

Substitute x = sin(t):

∫sin²(t)*cos(t)/cos(t) dt from 0 to pi/4

1/2*∫(1 - cos(2t)) dt from 0 to pi/4

1/2*[t - sin(t)*cos(t) eval. from 0 to pi/4]

1/2*[pi/4 - 1/2] = pi/8 - 1/4

http://calc101.com/trig_sub_4.html
this may help

1/8 (-2 + pi)