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 8.6 - Integral Tables and Computer Algebra Systems - Exercises 8.6 - Page 481: 54


$2\sqrt {3} $

Work Step by Step

Suppose $\sin u=y\implies dy=\cos udu$ Now the limits become: $y=0$, $u=0$ and for $y=\sqrt3/2$, $u=\pi/3$ Let $I=\int^{\pi/3}_0\dfrac{\cos udu}{(1-\sin^2u)^{5/2}} \\=\int^{\pi/3}_0\dfrac{\cos udu}{(\cos^2u)^{5/2}} \\=\int^{\pi/3}_0\sec^4udu \\=[\dfrac{\sec^2u\tan u}{3}\Big]^{\pi/3}_0+(2/3) \space \times \int^{\pi/3}_0\sec^2(u) du\\=(1/3) \times (4\sqrt3)+(2/3) \times(\sqrt3-0) \\=\dfrac{4\sqrt3}{3}+\dfrac{2\sqrt3}{3} \\=2\sqrt {3} $
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