Answer
$$\frac{1}{3}$$
Work Step by Step
$$\eqalign{
& \int_0^1 {x\sqrt {1 - {x^2}} } dx \cr
& {\text{substitute }}u = 1 - {x^2},{\text{ }}du = - 2xdx \cr
& {\text{express the limits in terms of }}u \cr
& x = 0{\text{ implies }}u = 1 - {\left( 0 \right)^2} = 1 \cr
& x = 1{\text{ implies }}u = 1 - {\left( 1 \right)^2} = 0 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_0^1 {x\sqrt {1 - {x^2}} } dx = - \frac{1}{2}\int_1^0 {{u^{1/2}}du} \cr
& {\text{find the antiderivative}} \cr
& = - \frac{1}{2}\left. {\left( {\frac{{{u^{3/2}}}}{{3/2}}} \right)} \right|_1^0 \cr
& {\text{use the fundamental theorem}} \cr
& = - \frac{1}{3}\left( {{0^{1/2}} - {1^{1/2}}} \right) \cr
& {\text{simplify}} \cr
& = \frac{1}{3} \cr} $$
