Question

$intlimits^1_0 { frac{ sqrt{x} }{1+x} } , dx$

Answer 1

$LARGE int_{0}^{1}{sqrt{x}over 1+x}{mathrm{d}x}=left [ u^2=x, {mathrm{d}x}=2u{mathrm{d}u} right ]=2int_{0}^{1}{u^2over 1+u}{mathrm{d}u} = 2left ( int_{0}^{1}{u^2+1-1over 1+u^2}{mathrm{d}u} right )=2left (int_{0}^{1} {mathrm{d}u}-int_{0}^{1}{1over 1+u^2}{mathrm{d}u} right )=2u|_{0}^{1}-2arctgu|_{0}^{1}=2-{pi over 2}$