## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 5: Integrals - Practice Exercises - Page 308: 58

#### Answer

$$2$$

#### Work Step by Step

\eqalign{ & \int_{\pi /4}^{3\pi /4} {{{\csc }^2}x} dx \cr & {\text{Recall that }}\cr &\frac{d}{{dx}}\left[ {\cot x} \right] = - {\csc ^2}x\cr &{\text{ Then}}{\text{,}} \cr & \int_{\pi /4}^{3\pi /4} {{{\csc }^2}x} dx = - \left( {\cot x} \right)_{\pi /4}^{3\pi /4} \cr & {\text{Evaluate}} \cr & = - \left( {\cot \left( {\frac{{3\pi }}{4}} \right) - \cot \left( {\frac{\pi }{4}} \right)} \right) \cr & {\text{Simplify}} \cr & = - \left( { - 1 - 1} \right) \cr & = 2 \cr}

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