Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 13 - The Trigonometric Functions - Chapter Review - Review Exercises - Page 702: 78

Answer

$$\frac{{2{{\left( {\sin x} \right)}^{5/2}}}}{5} + C$$

Work Step by Step

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