Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 39

Answer

$$ - \cot x + \tan x + C$$

Work Step by Step

$$\eqalign{ & \int {\frac{{{{\csc }^4}x}}{{{{\cot }^2}x}}dx} \cr & {\text{split off }}{\csc ^4}x \cr & = \int {\frac{{{{\csc }^2}x}}{{{{\cot }^2}x}}\left( {{{\csc }^2}x} \right)dx} \cr & {\text{pythagorean identity}} \cr & = \int {\frac{{{{\cot }^2}x + 1}}{{{{\cot }^2}x}}\left( {{{\csc }^2}x} \right)dx} \cr & = \int {\left( {1 + \frac{1}{{{{\cot }^2}x}}} \right)\left( {{{\csc }^2}x} \right)dx} \cr & u = \cot x,{\text{ }}du = - {\csc ^2}xdx \cr & = - \int {\left( {1 + \frac{1}{{{u^2}}}} \right)du} \cr & {\text{evaluate}} \cr & = - u + \frac{1}{u} + C \cr & {\text{replace }}u = \cot x \cr & = - \cot x + \frac{1}{{\cot x}} + C \cr & = - \cot x + \tan x + C \cr} $$
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