Calculus 8th Edition

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

Chapter 4 - Integrals - 4.4 Indefinite Integrals and the Net Change Theorem - 4.4 Exercises - Page 336: 15


$-cot(t) + cos(t) + C$

Work Step by Step

We can simplify the fraction in the integrand as $\int \frac{1}{sin^2(t)} - sin(t) dt = \int csc^2(t) - sin(t) dt$, by the trigonometric identity $ \frac{1}{sin(t)} = csc(t)$ Evaluating the integrand gives us $-cot(t) +cos(t) + C$ Note: Recall the trigonometric derivative identities $$ \frac{d}{dt} cot(t) = -csc^2(t) $$ and $$ \frac{d}{dt} cos(t) = -sin(t) $$
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