Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 31



Work Step by Step

Given $$\int \frac{d x}{x^{2}+2 x+5}$$ \begin{aligned} \int \frac{1}{x^{2}+2 x+5} d x &=\int \frac{1}{(x+1)^{2}+4} d x \end{aligned} Let $$x+1=2\tan u,\ \ \ \ \ \ \ \ dx=2\sec^2 udu $$ Then \begin{align*} \int \frac{1}{x^{2}+2 x+5} d x &=\int \frac{2\sec^2 udu }{4\tan^{2}u+4} \\ &=\int \frac{2\sec^2 udu }{4\sec^{2}u } \\ &=\frac{1}{2}\int du\\ &=\frac{1}{2}u+C\\ &=\frac{1}{2}\tan^{-1}\frac{x+1}{2}+C\\ \end{align*}
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