Calculus (3rd Edition)

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

Chapter 5 - The Integral - Chapter Review Exercises - Page 279: 79


$$\int_{-8}^{8} \frac{x^{15} d x}{3+\cos ^{2} x}=0$$

Work Step by Step

For the integral $\int_{-8}^{8} \frac{x^{15} d x}{3+\cos ^{2} x}$, we have $$f(-x)=\frac{(-x)^{15} }{3+\cos ^{2}(-x)}=-\frac{x^{15} d x}{3+\cos ^{2} x}=-f(x).$$ Hence, $f(x)$ is an odd function and then $\int_{-a}^af(x)dx=0$. That is, $$\int_{-8}^{8} \frac{x^{15} d x}{3+\cos ^{2} x}=0$$
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