Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 459: 4


$$-\frac{12 x+1}{144} e^{-12 x}+C$$

Work Step by Step

Given $$ \int x e^{-12 x} d x$$ Let \begin{align*} u&=x\ \ \ \ \ \ \ dv =e^{-12x}dx\\ du&=dx\ \ \ \ \ v=\frac{-1}{12}e^{-12x} \end{align*} Then \begin{aligned} \int x e^{-12 x} d x &=x \cdot\left(-\frac{e^{-12 x}}{12}\right)-\int\left(-\frac{e^{-12 x}}{12}\right) \cdot 1 d x \\ &=-\frac{1}{12} x e^{-12 x}+\frac{1}{12} \int e^{-12 x} d x \\ &=-\frac{1}{12} x e^{-12 x}+\frac{1}{12} \cdot\left(-\frac{e^{-12 x}}{12}\right)+C \\ &=-\frac{1}{12} x e^{-12 x}-\frac{1}{144} e^{-12 x}+C \\ &=-\frac{12 x+1}{144} e^{-12 x}+C \end{aligned}
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