Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Section 8.1 - Using Basic Integration Formulas - Exercises 8.1 - Page 448: 8

Answer

\begin{aligned} \int \frac{2^{\ln z^{3}} d z}{16 z} = \frac{ 1}{16 \ln 8}8^{\ln z}+c\\ \end{aligned}

Work Step by Step

Given $$\int \frac{2^{\ln z^{3}} d z}{16 z}$$ So, we get \begin{aligned} I&=\int \frac{2^{\ln z^{3}} d z}{16 z}\\ &=\int \frac{2^{3\ln z } d z}{16 z}\\ &=\int \frac{8^{\ln z } d z}{16 z}\\ \end{aligned} Let $$ y=8^{\ln z } \Rightarrow dy=\frac{\ln 8}{z}8^{\ln z }dz \Rightarrow \frac{1}{z}8^{\ln z }dz=\frac{1}{\ln 8}dy $$ So, we get \begin{aligned} I&=\int \frac{8^{\ln z } d z}{16 z}\\ &=\int \frac{ dy}{16 \ln 8}\\ &= \frac{ 1}{16 \ln 8} \int dy\\ &= \frac{ 1}{16 \ln 8}y+c\\ &= \frac{ 1}{16 \ln 8}8^{\ln z}+c\\ \end{aligned}
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