Calculate: integral from 0 to 2 of x/(sqrt(4-x^2))

Expression: $\int _{0}^{2}\frac{x}{\sqrt{4-x^{2}}}dx$

Apply u-substitution

$=\int _{4}^{0}-\frac{1}{2\sqrt{u}}du$

$\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx, a

$=-\int _{0}^{4}-\frac{1}{2\sqrt{u}}du$

Take the constant out $\int{a\cdot{f(x)}dx}=a\cdot\int{f(x)dx}$

$=-(-\frac{1}{2}\cdot \int _{0}^{4}\frac{1}{\sqrt{u}}du)$

Apply the Power Rule

$=-(-\frac{1}{2}[2u^{\frac{1}{2}}]_{0}^{4})$

Simplify $-(-\frac{1}{2}[2u^{\frac{1}{2}}]_{0}^{4}):{\quad}\frac{1}{2}[2\sqrt{u}]_{0}^{4}$

$=\frac{1}{2}[2\sqrt{u}]_{0}^{4}$

Compute the boundaries $:{\quad}4$

$=\frac{1}{2}\cdot 4$

Simplify

$=2$