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Chapter 7: Problem 22

Evaluate the integrals. $$\int e^{\csc (\pi+\pi)} \csc (\pi+t) \cot (\pi+t) d t$$

Short Answer

Expert verified
The integral evaluates to zero plus a constant of integration: \\(C\\).

Step by step solution

01

Simplify the Expression Inside the Integral

First, simplify the integrand by recognizing any constants within it. Notice that \( e^{\csc(\pi+\pi)} \) can be simplified. Given \( \sin(\pi + \pi) = 0 \), the cosecant becomes undefined, making \( e^{\csc(\pi + \pi)} \) a constant provided by the problem's context as simplified to zero.
02

Re-examine the Integral's Simplification

Given the correction of \( e^{\csc(\pi+\pi)} \), noting it represents an undefined expression, the problem context simplifies it to form a trivial integrand: \(\int 0 \, dt\). This reflects a constant integrand effectively equal to zero.
03

Evaluate the Simplified Integral

The integral of 0 with respect to \( t \) over any interval equals zero since the area under the curve of a function equal to zero is zero. Thus, \(\int 0 \, dt = C\), 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.

Trigonometric Integrals
Trigonometric integrals are a fundamental part of integral calculus. Their essential purpose is to solve integrals that involve trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant. These integrals appear frequently in engineering, physics, and other scientific fields.

When dealing with trigonometric integrals, we often use identities and algebraic manipulation to simplify the expression. This approach helps in making the integral more manageable and easier to solve.

  • The use of trigonometric identities is key. For instance, the identities for double angles or Pythagorean identities can transform the integral into a form that is easier to handle.
  • Simplifying expressions sometimes results directly in zero, as seen in some integrals where trigonometric functions become undefined or cancel out.
  • It's crucial to recognize when an expression simplifies to a known result such as zero or a constant, which can drastically change the needed computation.
Understanding these integral simplification techniques is crucial for successfully evaluating more complex integrals.
Cosecant Function
The cosecant function, denoted as \(\csc(\theta)\), is the reciprocal of the sine function. Therefore, \(\csc(\theta) = \frac{1}{\sin(\theta)}\). This function is critical when dealing with integrals involving trigonometric relationships.

In the integral provided, \(\csc(\pi+t)\) appears frequently. Understanding the behavior of this function helps in predicting when expressions can become undefined or when simplifications are possible.

  • The cosecant function becomes undefined at angles where the sine value is zero, such as at 0, \(\pi\), 2\(\pi\), etc.
  • Recognizing these points is vital as they can influence the region of integration or lead to simplifying the integral expression.
  • When simplifying trigonometric expressions, always evaluate the sine portion to check for undefined behavior.
This comprehension is necessary for both handling the function's properties and avoiding errors during integral evaluation.
Constant of Integration
A constant of integration, commonly denoted as \(C\), appears whenever we perform an indefinite integral. It reflects that there are infinitely many antiderivatives of a function, differing only by a constant.

In the given solution, after simplifying the integral to zero, we stated the result as \(\int 0\, dt = C\). This arises because the indefinite integral does not produce a specific value without initial conditions or limits.

  • The constant of integration is crucial in expressing the general solution of an indefinite integral.
  • When evaluating indefinite integrals, always include \(C\) in your solution to account for all possible antiderivatives.
  • In applied contexts, initial conditions may define \(C\), turning the solution into a more specific, exact antiderivative.
Understanding the role of the constant of integration ensures accuracy and completeness in solutions.

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