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Chapter 8: Problem 31

Find the indefinite integral. $$ \int \csc \pi x \cot \pi x d x $$

Short Answer

Expert verified
The indefinite integral of \(\int \csc \pi x \cot \pi x \, dx\) is \(-\frac{1}{\pi} \csc(\pi x) + C\), where C represents the constant of integration.

Step by step solution

01

Identify suitable substitution

Let the substitution variable u be defined as \(u = \pi x\). The differential du then is \(du = \pi dx\) or \(dx = \frac{1}{\pi}du\). Consequently, the integral \(\int \csc \pi x \cot \pi x \, dx\) transforms to \[ \int \csc u \cot u \, \frac{1}{\pi} du \].
02

Rewrite in terms of standard trigonometric function

\(\csc u \cot u\) is the derivative of \(-\csc u\), which can be rewritten as \(-\frac{d}{du}(\csc u)\). So, the transformed integral becomes \[ -\frac{1}{\pi} \int \frac{d}{du}(\csc u) \, du \].
03

Apply integral rules

From the rule of integration, the integral of a derivative function \(\int \frac{d}{du}(f(u)) \, du\) is simply the function itself \(f(u)\). Therefore, applying this rule will give us the integral of \(\frac{d}{du}(\csc u)\) as \(-\csc u\)
04

Back substitution

The variable u was earlier defined as \(\pi x\), substituting back will yield the final result to the problem \[-\frac{1}{\pi} \csc(\pi x) + C\] where C is the constant of integration.

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Key Concepts

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

Integration by Substitution
Integration by substitution is a powerful technique used to solve integrals, especially when the function looks complex at first glance. It involves changing variables to simplify the integral:
  • Identify a substitution that makes the integral easier to evaluate, typically by setting a variable like \(u\) to a part of the original function, such as \(u = \pi x\).
  • Find the differential, \(du\), in terms of the original variable, to transform the integral fully. This involves determining \(du = \pi \, dx\), which transforms \(dx = \frac{1}{\pi} du\).
  • Rewrite the integral using the substitution, which changes the variables and usually simplifies the function.
In this exercise, the substitution \(u = \pi x\) simplifies the problem by changing the integral into terms based on \(u\), making it easier to solve.
Trigonometric Integrals
Trigonometric integrals often involve products of trigonometric functions, which can be intricate to solve. Recognizing standard derivatives and integrals involving these functions is crucial:
  • The integral \(\int \csc u \cot u \, du\) is directly linked to the derivative \(-\csc u\). This is an instance where recognizing the derivative form simplifies the integration process.
  • This integral becomes a straightforward task once it is realized that \(\csc u \cot u\) is the derivative for \(-\csc u\), allowing a direct application of basic integral rules.
In this example, by recognizing this relationship, we easily transition to solving the integral \(-\frac{1}{\pi} \int \frac{d}{du}(\csc u) \, du\), leading to a smooth solution.
Integration Rules
Various integration rules simplify the process, especially when dealing with the derivatives of known functions:
  • The integral of a derivative function \(\int \frac{d}{du}(f(u)) \, du\) equals the function \(f(u)\) itself. This powerful rule allows us to directly integrate derivative expressions.
  • Constants like \(-\frac{1}{\pi}\) can be factored out of the integral, simplifying the manipulation of the expression.
  • Remembering to back-substitute the original variable at the end ensures that the solution corresponds to the problem's initial terms.
Here, using these rules, the integral \(-\csc u\) easily converts back to \(-\csc(\pi x)\) after substituting \(u = \pi x\). Always add the constant \(C\) for indefinite integrals to account for all possible antiderivatives.

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Most popular questions from this chapter

For what value of \(c\) does the integral \(\int_{1}^{\infty}\left(\frac{c x}{x^{2}+2}-\frac{1}{3 x}\right) d x\) converge? Evaluate the integral for this value of \(c\).

The Gamma Function \(\Gamma(n)\) is defined by $$ \Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, \quad n>0. $$ (a) Find \(\Gamma(1), \Gamma(2)\), and \(\Gamma(3)\). (b) Use integration by parts to show that \(\Gamma(n+1)=n \Gamma(n)\). (c) Write \(\Gamma(n)\) using factorial notation where \(n\) is a positive integer.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. \(y=\cos \frac{x}{2}, \quad y=\sin \frac{x}{2}, \quad x=0, \quad x=\pi / 2\)

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by$$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$if the improper integral exists. Laplace Transforms are used to solve differential equations, find the Laplace Transform of the function. $$ f(t)=\cosh a t $$

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by$$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$if the improper integral exists. Laplace Transforms are used to solve differential equations, find the Laplace Transform of the function. $$ f(t)=1 $$
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