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 395: 49

Answer

$$\ln 4-\frac{3}{4}$$

Work Step by Step

Given $$\int_{1}^{2} x \ln x d x$$ Let \begin{align*} u&=\ln x\ \ \ \ \ \ \ \ dv= xdx\\ du&= \frac{ 1}{x} dx\ \ \ \ \ \ \ \ v=\frac{1}{2}x^2 \end{align*} Then \begin{align*} \int_{1}^{2} x \ln x d x&= \frac{1}{2}x^2 \ln x\bigg|_{1}^{2} -\frac{1}{2} \int_{1}^{2} x d x \\ &= \frac{1}{2}x^2 \ln x\bigg|_{1}^{2} -\frac{1}{4} x^2\bigg|_{1}^{2} \\ &=2\ln 2-\frac{3}{4} \\ &= \ln 4-\frac{3}{4} \end{align*}
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