Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises - Page 346: 10


$-\frac{2}{3}(1+cos(t))^{3/2} +C$

Work Step by Step

Evaluate the Integral using substitution: $\int sin(t)\sqrt{1+cos(t)}dt$ Substitution Rule: $\int f(g(x))gā€™(x)dx = \int f(u)du$ $u= 1+cos(t)$ $du =-sin(t)$ In this expression $du$ is multiplied by $-1$ to be $sin(t)$ Solve the integral in terms of $u$: $\int \sqrt{u}(-1)du$ $(-1)\int u^{1/2}du$ $(-1) \frac{2u^{3/2}}{3} +C$ Substitute for $u$: $-\frac{2}{3}(1+cos(t))^{3/2} +C$
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