Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 4 - Applications of the Derivative - 4.9 Antiderivatives - 4.9 Exercises: 29

Answer

$$\frac{9}{4}{x^{4/3}} + 6{x^{2/3}} + 6x + C$$

Work Step by Step

$$\eqalign{ & \int {\left( {3{x^{1/3}} + 4{x^{ - 1/3}} + 6} \right)} dx \cr & {\text{use power rule for indefinite integrals}} \cr & = 3\left( {\frac{{{x^{4/3}}}}{{4/3}}} \right) + 4\left( {\frac{{{x^{2/3}}}}{{2/3}}} \right) + 6x + C \cr & {\text{simplify}} \cr & = \frac{9}{4}{x^{4/3}} + 6{x^{2/3}} + 6x + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dx}}\left( {\frac{9}{4}{x^{4/3}} + 6{x^{2/3}} + 6x + C} \right) \cr & = \frac{9}{4}\left( {\frac{4}{3}} \right){x^{1/3}} + 6\left( {\frac{2}{3}} \right){x^{ - 1/3}} + 6\left( 1 \right) + 0 \cr & = 3{x^{1/3}} + 4{x^{ - 1/3}} + 6 \cr} $$
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