## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 45

#### Answer

$$x\ln (x^2+9)-2 \left(x-3\tan^{-1}(x/3) \right)+C$$

#### Work Step by Step

Given $$\int \ln (x^2+9) d x$$ Let \begin{align*} u&=\ln (x^2+9)\ \ \ \ \ \ \ \ \ \ dv= xdx\\ du&=\frac{2x}{x^2+9}dx\ \ \ \ \ \ \ \ v= x \end{align*} Then \begin{align*} \int \ln (x^2+9) d x&=x\ln (x^2+9)-2\int \frac{x^2}{x^2+9}dx\\ &=x\ln (x^2+9)-2\int \frac{x^2+9-9}{x^2+9}dx\\ &= x\ln (x^2+9)-2\int \left(1- \frac{9}{x^2+9}\right)dx\\ &= x\ln (x^2+9)-2 \left(x-3\tan^{-1}(x/3) \right)+C \end{align*}

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