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 - Practice Exercises - Page 517: 4


$(x) \cos^{-1}(\dfrac{x}{2})-\sqrt {4-x^2}+c$

Work Step by Step

The integration by parts formula suggests that $\int a'(x) b(x)=a(x) b(x)-\int a(x) b'(x)dx$ Now, we have $I=\int \cos^{-1}(\dfrac{x}{2})dx= \cos^{-1}(\dfrac{x}{2})\int dx-\int (\dfrac{x}{2\sqrt {1-(\dfrac{x}{2})^2}} dx$ or, $I=\cos^{-1}(\dfrac{x}{2})\int dx-\int (\dfrac{x}{2\sqrt {1-(\dfrac{x}{2})^2}} dx$ or, $I=(x) \cos^{-1}(\dfrac{x}{2})\int \dfrac{x}{\sqrt {4-x^2}} dx$ consider $p=4-x^2$ or, $-\dfrac{dp}{2}=x dx$ Thus, $I=x \cos^{-1}(\dfrac{x}{2})-\int \dfrac{da}{2 \sqrt a}$ or, $I=(x) \cos^{-1}(\dfrac{x}{2})-\sqrt {4-x^2}+c$
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