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Chapter 8: Problem 5

Show that \(\int \cot (x) \mathrm{d} x=\ln |\sin (x)|+C\)

Short Answer

Expert verified
By performing a substitution \(u = \sin(x)\), the integral \(\text{cot}(x)\text{d}x\) simplifies to \(\text{ln}|\text{sin}(x)| + C\).

Step by step solution

01

Rewrite the Integral Using Trigonometric Identity

Rewrite \(\text{cot}(x)\) as \(\frac{\cos(x)}{\sin(x)}\). Thus, the integral becomes \(\int \cot(x) \mathrm{d}x = \int \frac{\cos(x)}{\sin(x)} \mathrm{d}x\).
02

Perform a Substitution

Let \(u = \sin(x)\). Then, \(\frac{d}{dx} (u) = \cos(x)\) which implies \(du = \cos(x) dx\).
03

Substitute and Integrate

Substitute \(u\) into the integral: \(\int \frac{\cos(x)}{\sin(x)} \mathrm{d}x = \int \frac{1}{u} \mathrm{d}u\). The integral of \(\frac{1}{u}\) is \(\ln|u| + C\).
04

Re-substitute \(\text{sin}(x)\)

Since \(u = \sin(x)\), re-substitute \(u\) back to get \(\text{ln}|\sin(x)| + C\). Therefore, \(\int \cot(x) \mathrm{d}x = \ln| \sin(x) | + C\).\

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Substitution
Trigonometric substitution is a technique used in integration to simplify expressions involving roots, squares, and other forms where standard algebraic methods can be challenging. In the problem \(\text{Show that \(\int \cot (x)\d x=\ln \|\sin (x)\|+C\)\)}, we use trigonometric substitution to rewrite \(\text{cot}(x)\) as \(\frac{\cos(x)}{\sin(x)}\).
This substitution simplifies the integral and makes it easier to handle through further algebraic manipulations.
Indefinite Integrals
Indefinite integrals represent the family of all functions that, when differentiated, yield the integrand. For instance, if \(\frac{\mathrm{d}}{\mathrm{d}x}F(x) = f(x)\), then the indefinite integral of \(\text{f(x)}\) is \(\text{F(x) + C}\), where \(\text{C}\) is the constant of integration.
In the given exercise, the indefinite integral \(\int \cot(x)\mathrm{d}x\) is found to be \(\ln|\text{sin}(x)\| + C\).
This is done by recognizing the integral's form and performing specific substitution steps.
Trigonometric Identities
Trigonometric identities are important tools in calculus, as they help in simplifying integrals and derivatives involving trigonometric functions. In our problem, we use the identity \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\) to convert the integral to a more workable form: \(\int \frac{\cos(x)}{\sin(x)} \mathrm{d}x\).
This transformation makes the integral easier to approach with techniques like substitution and further simplification.
U-substitution
U-substitution is a method used to simplify integration by substituting a part of the integrand. In this exercise, we set \(\text{u} = \text{sin}(x)\) and recognized that \(\frac{\mathrm{d}u}{\mathrm{d}x} = \text{cos}(x)\), which implies \(\text{du} = \text{cos}(x) \mathrm{d}x\).
This allows us to rewrite the integral \(\int \frac{\text{cos}(x)}{\text{sin}(x)} \mathrm{d}x\) as \(\int \frac{1}{u} \mathrm{d}u\).
The integral of \(\frac{1}{u}\) is \(\text{ln}|u| + C\), and replacing \(\text{u}\) back with \(\text{sin}(x)\) gives the final answer: \text{ln}|\text{sin}(x)| + C\.

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