Answer
$\frac{{\pi}^{3/2}}{3\sqrt 2} - 3$
Work Step by Step
$$\eqalign{
&\text{Let }I= \int_0^{\pi /2} {\left( {\sqrt t - 3\cos t} \right)} dt \cr
& = \int_0^{\pi /2} {\left( {{t^{1/2}} - 3\cos t} \right)} dt \cr
& {\text{Integrate by the power rule and }}\int {\cos x} dx = \sin x + C \cr
&I = \left[ {\frac{{{t^{3/2}}}}{{3/2}} - 3\sin t} \right]_0^{\pi /2} \cr
& = \left[ {\frac{{2{t^{3/2}}}}{3} - 3\sin t} \right]_0^{\pi /2} \cr
& {\text{Using the fundamental theorem of calculus}}{\text{, part 2}} \cr
& I = \left[ {\frac{{2{{\left( {\pi /2} \right)}^{3/2}}}}{3} - 3\sin \left( {\frac{\pi }{2}} \right)} \right] - \left[ {\frac{{2{{\left( 0 \right)}^{3/2}}}}{3} - 3\sin \left( 0 \right)} \right] \cr
& {\text{Simplify}} \cr
& I= \left[ {\frac{{2{{\left( {\pi /2} \right)}^{3/2}}}}{3} - 3\left( 1 \right)} \right] - \left[ 0 \right] \cr
& = \frac{{2{{\left( {\pi /2} \right)}^{3/2}}}}{3} - 3 \cr
& = \frac{{\pi}^{3/2}}{3\sqrt 2} - 3 \cr} $$