Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.1 Integration by Parts - Exercises - Page 396: 78


$$f(x)=\ln x+C$$

Work Step by Step

Given $$ \int f(x) e^{x} d x=f(x) e^{x}-\int x^{-1} e^{x} d x$$ Let \begin{align*} u&= f(x) \ \ \ \ \ \ \ \ \ \ dv= e^xdx\\ du&=f'(x) dx\ \ \ \ \ \ v=e^x \end{align*} Then \begin{align*} \int f(x) e^{x} d x&= f(x) e^x -\int f'(x)e^xdx\\ f(x) e^{x}-\int x^{-1} e^{x} d x&= f(x) e^x -\int f'(x)e^xdx \end{align*} By comparison \begin{align*} \int x^{-1} e^{x} d x&= \int f'(x)e^xdx\\ \int x^{-1} d x&= \int f'(x) dx\\ \ln x+C&=f(x) \end{align*}
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