Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 5 - Review - Exercises - Page 424: 69

Answer

$2 \int_{a}^{b} f(x)f'(x)dx = [f(b)]^2-[f(a)]^2$

Work Step by Step

$2 \int_{a}^{b} f(x)f'(x)dx$ Let $u = f(x)$ $\frac{du}{dx} = f'(x)$ $dx = \frac{du}{f'(x)}$ When $x = a$, then $u = f(a)$ When $x = b$, then $u = f(b)$ $2 \int_{f(a)}^{f(b)} uf'(x)\cdot \frac{du}{f'(x)}$ $=2 \int_{f(a)}^{f(b)} u~du$ $=2 (\frac{1}{2}u^2)~\Big\vert_{f(a)}^{f(b)}$ $=(u^2)~\Big\vert_{f(a)}^{f(b)}$ $=[f(b)]^2-[f(a)]^2$
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