Answer
$$2$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {5{{\left( {\sin x} \right)}^{3/2}}\cos x} dx \cr
& {\text{Use the substitution method}}{\text{:}} \cr
& u = \sin x,\,\,\,\,du = \cos xdx,\,\,\,\,\,dx = \frac{{du}}{{\cos x}} \cr
& {\text{Then}}{\text{,}} \cr
& \int {5{{\left( {\sin x} \right)}^{3/2}}\cos x} dx = \int {5{u^{3/2}}\cos x} \left( {\frac{{du}}{{\cos x}}} \right) \cr
& = \int {5{u^{3/2}}du} \cr
& {\text{Integrate by the power rule}} \cr
& = 5\left( {\frac{{{u^{5/2}}}}{{5/2}}} \right) + C \cr
& = 2{u^{5/2}} + C \cr
& {\text{Write in terms of }}x;{\text{ replace }}u = \sin x \cr
& = 2{\left( {\sin x} \right)^{5/2}} + C \cr
& \int_0^{\pi /2} {5{{\left( {\sin x} \right)}^{3/2}}\cos x} dx = \left( {2{{\left( {\sin x} \right)}^{5/2}}} \right)_0^{\pi /2} \cr
& {\text{Evaluate}} \cr
& = 2{\left( {\sin \frac{\pi }{2}} \right)^{5/2}} - 2{\left( {\sin 0} \right)^{5/2}} \cr
& = 2{\left( 1 \right)^{5/2}} - 2{\left( 0 \right)^{5/2}} \cr
& = 2 \cr} $$
