Answer
$$\frac{1}{3}$$
$$Or$$
$$0.3333333$$
Work Step by Step
$$\int_{0}^{1} x \sqrt{1-x^2} dx$$
$u=1-x^2$
$du=-2xdx$
$-\frac{1}{2}du=xdx$
$$-\frac{1}{2} \int_{0}^{1} \sqrt u du$$
$-\frac{1}{2}* _{0}^{1} |\frac{2}{3} u^{\frac{3}{2}}$
$-\frac{1}{3}*_{0}^{1}|(1-x^2)^{\frac{3}{2}}$
$-\frac{1}{3} (0-1)$
$$=\frac{1}{3}=0.3333333$$
