/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Find the indefinite integral. ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 27

Find the indefinite integral. $$ \int \frac{\csc ^{2} x}{\cot ^{3} x} d x $$

Short Answer

Expert verified
\(-\frac{1}{2 \cot^{2}x} - \ln |\cot x|+C\)

Step by step solution

01

Express Trigonometric Function Using Pythagorean Identity

Express \(\csc^2(x)\) as \(1+\cot^2(x)\) using the Pythagorean identity. We get \(\int \frac{1+\cot^2(x)}{\cot ^{3} x} dx\)
02

Simplify The Integral

Divide through the fraction, which leads to \(\int (\frac{1}{\cot^{3}x} + \frac{1}{\cot{x}})dx= \int \cot^{-3}x + \cot^{-1}x dx\)
03

Simplify Using Substitution

Let \(u=\cot(x)\), then \(-du = \csc^{2}(x) dx\). Therefore the integral \(\int \cot^{-3}x dx= \int u^{-3} du\) and \(\int \cot^{-1}x dx = \int u^{-1} du\). Taking into account for the extra negative in the du we then have \(\int - u^{-3} du - \int u^{-1} du\).
04

Integrate

Each of these integrals can be integrated using basic rules of integration which leads to \(-\frac{1}{2} u^{-2} - \ln |u|\)
05

Back Substitute

Remembering that \(u = \cot(x)\), substitute back into the expression and simplify, leading to \(-\frac{1}{2 \cot^{2}x} - \ln |\cot x|\)
06

Express Answer in Standard Form

We generally express the result of an integral in the standard form which is the antiderivative plus a constant C. So, write the final answer as \(-\frac{1}{2 \cot^{2}x} - \ln |\cot x|+C\)

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Most popular questions from this chapter

use a graphing utility to graph the function \(f\) and find \(\lim _{x \rightarrow 0} f(x)\). $$ f(x)=\frac{\tan 2 x}{3 x} $$

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