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Let's use trig substitution to solve a problem.
∫ x² / (√ 1 - x²) dx
Let x = sin θ. dx = cos θ dθ. Substituting
∫ sin² θ cos θ / √ 1 - sin² θ dθ
= ∫ sin² θ cos θ / √ cos² θ dθ
,= ∫ sin² θ cos θ / cos θ dθ. ( cos θ cancels out )
= ∫ sin² θ dθ. { sin² θ =1/2 ( 1 - cos 2θ ) }
= 1/2 ∫ (1 - cos 2θ ) dθ
= 1/2 ∫ dθ - ∫ cos 2θ dθ
= 1/2 ( θ - 1/2 sin 2θ ) + C
= 1/2 [ arc sin x - 1/2 ( 2 sin θ cos θ ) ] +C
= 1/2 arc sin x - 1/2 x ( √1 - x² ) + C
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