## Calculus with Applications (10th Edition)

$$\frac{{2{{\left( {\sin x} \right)}^{5/2}}}}{5} + C$$
\eqalign{ & \int {{{\left( {\sin x} \right)}^{3/2}}} \cos xdx \cr & {\text{set }}u = \sin x{\text{ then }}\frac{{du}}{{dx}} = \cos x,\,\,\,\,\,\,\,\,\frac{{du}}{{\cos x}} = dx \cr & {\text{write the integrand in terms of }}u \cr & \int {{{\left( {\sin x} \right)}^{3/2}}\cos x} dx = \int {{u^{3/2}}\cos x} \left( {\frac{{du}}{{\cos x}}} \right) \cr & = \int {{u^{3/2}}} du \cr & {\text{integrate by using the power rule for integration}} \cr & = \frac{{{u^{5/2}}}}{{5/2}} + C \cr & = \frac{{2{u^{5/2}}}}{5} + C \cr & {\text{write in terms of }}x \cr & = \frac{{2{{\left( {\sin x} \right)}^{5/2}}}}{5} + C \cr}