Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.1 Integration by Parts - Exercises - Page 395: 26


$$\int arcsin\,x\,dx=x\,arcsin\,x+\sqrt{1-x^{2}}+C$$

Work Step by Step

${(x\,arcsin\,x)}'=arcsin\,x+\frac{x}{\sqrt{1-x^{2}}}$ $$\int arcsin\,x\,dx=x\,arcsin\,x- \int\frac{x}{\sqrt{1-x^{2}}}dx$$ $$=x\,arcsin\,x+\frac{1}{2}\int (1-x^{2})^{-\frac{1}{2}}d(1-x^{2})$$ $$=x\,arcsin\,x+(1-x^{2})^{\frac{1}{2}}+C$$ $$=x\,arcsin\,x+\sqrt{1-x^{2}}+C$$
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