Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.1 Exercises: 69

Answer

The integral is: $$\frac{1}{3}\arctan\frac{x+2}{3}+c$$ where $c$ is arbitratry. The graph of the antiderivatives is on the following figure.
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Work Step by Step

Using Wolfram Mathematica (which is an example of computer algebra system) with the code y = Integrate[1/(x^2 + 4 x + 13), x] Plot[{y + 2, y - 1}, {x, -10, 10}] We get that integral $$\int\frac{1}{x^2+4x+13} dx=\frac{1}{3}\arctan\frac{x+2}{3}+c.$$ We get two antiderivatives for two different choices for $c$. We have chosen $c=2$ (blue) and $c=-1$ (orange) and they are plotted on the graph.
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