Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 4 - Integration - 4.2 The Indefinite Integral - Exercises Set 4.2 - Page 279: 16

Answer

$$4y + \frac{{4{y^3}}}{3} + \frac{{{y^5}}}{5} + C$$

Work Step by Step

$$\eqalign{ & \int {{{\left( {2 + {y^2}} \right)}^2}dy} \cr & {\text{expand integrand}} \cr & = \int {\left( {{2^2} + 2\left( 2 \right)\left( {{y^2}} \right) + {{\left( {{y^2}} \right)}^2}} \right)dy} \cr & = \int {\left( {4 + 4{y^2} + {y^4}} \right)dy} \cr & = \int {4dy} + \int {4{y^2}} dy + \int {{y^4}} dy \cr & {\text{power rule}} \cr & = 4y + \frac{{4{y^{2 + 1}}}}{{2 + 1}} + \frac{{{y^{4 + 1}}}}{{4 + 1}} + C \cr & = 4y + \frac{{4{y^3}}}{3} + \frac{{{y^5}}}{5} + C \cr & \cr & {\text{check by differentiation}} \cr & = \frac{d}{{dx}}\left[ {4y + \frac{{4{y^3}}}{3} + \frac{{{y^5}}}{5} + C} \right] \cr & = 4 + \frac{{4\left( {3{y^2}} \right)}}{3} + \frac{{5{y^4}}}{5} + C \cr & = 4 + 4{y^2} + {y^4} \cr & = {\left( {2 + y} \right)^2} \cr} $$
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