Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Section 8.8 - Improper Integrals - Exercises 8.8 - Page 501: 30



Work Step by Step

\begin{align*} \int_{2}^{4} \frac{d t}{t \sqrt{t^{2}-4}}&=\lim _{b \rightarrow 2^{+}}\int_{b}^{4} \frac{d t}{t \sqrt{t^{2}-4}}\\ &=\lim _{b \rightarrow 2^{+}}\left[\frac{1}{2} \sec ^{-1} \frac{t}{2}\right]\bigg|_{b}^{4}\\ &=\lim _{b \rightarrow 2^{+}}\left[\left(\frac{1}{2} \sec ^{-1} \frac{4}{2}\right)-\frac{1}{2} \sec ^{-1}\left(\frac{b}{2}\right)\right]\\ &=\frac{1}{2}\left(\frac{\pi}{3}\right)-\frac{1}{2} \cdot 0=\frac{\pi}{6} \end{align*}
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