Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition

Published by Brooks Cole
ISBN 10: 1-43904-957-2
ISBN 13: 978-1-43904-957-0

Chapter 5 - Accumulating Change: Limits of Sums and the Definite Integral - 5.9 Activities - Page 409: 4

Answer

$$\int 3(\ln 2) 2^{x}\left(1+2^{x}\right)^{3} d x =\frac{3}{4} \left(1+2^{x}\right)^4+c$$

Work Step by Step

Given $$ \int 3(\ln 2) 2^{x}\left(1+2^{x}\right)^{3} d x $$ Let $ u=1+2^{x}\ \to\ \ du=(\ln 2) 2^{x}dx$, so: \begin{align*} \int 3(\ln 2) 2^{x}\left(1+2^{x}\right)^{3} d x&=\int 3 u^{3} du\\ &=\frac{3}{4} u^4+c\\ &=\frac{3}{4} \left(1+2^{x}\right)^4+c \end{align*}
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