Calculus (3rd Edition)

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

Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 276: 51


$$\frac{2 \sqrt{\sin x+1}}{3}(\sin x-2)+C$$

Work Step by Step

Given $$\int \frac{\sin x \cdot \cos x}{\sqrt{\sin x+1}} d x$$ Let $$ u=\sin x+1\ \ \ \ \Rightarrow \ \ \ du = \cos xdx$$ Then \begin{aligned} \int \frac{\sin x \cdot \cos x}{\sqrt{\sin x+1}} d x &=\int \frac{u-1}{\sqrt{u}} d u \\ &=\int\left(u^{1 / 2}-u^{-1 / 2}\right) d u \\ &=\frac{2 u^{3 / 2}}{3}-2 \cdot u^{1 / 2}+C \\ &=\frac{\sqrt{\sin x+1}}{3}(2 \sin x-4)+C \\ &=\frac{2 \sqrt{\sin x+1}}{3}(\sin x-2)+C \end{aligned}
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