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int{x*sqrt(1-x^2)dx}
let u=1-x^2.....(1)
then,du/dx= -2x giving dx= -du/2x
int{x*sqrt(1-x^2)dx}
=int{x*sqrt(u)*(-du/2x)}
= (-1/2)*int(sqrt(u))du
= -(1/2)*u^(3/2)/(3/2) +C
= -(1/2)*(2/3)*u^(3/2)+C
= -(1/3)*(u)^(3/2)+C
but,from (1), u=1-x^2
substituting for u,
int{x*sqrt(1-x^2)dx} = -(1/3)*(1-x^2)^(3/2)
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Michael D. Holloway
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