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Chapter 4: Problem 44

In Exercises \(17-56,\) find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{1}{2}\left(\csc ^{2} x-\csc x \cot x\right) d x$$

Short Answer

Expert verified
The general antiderivative is \(-\frac{1}{2} \cot x + \frac{1}{2} \csc x + C\).

Step by step solution

01

Simplify the Integrand

The integrand is \( \frac{1}{2}(\csc^2 x - \csc x \cot x) \). Break it into two separate integrals: \( \frac{1}{2} \int \csc^2 x \, dx - \frac{1}{2} \int \csc x \cot x \, dx \).
02

Find Antiderivative for the First Term

The antiderivative of \( \csc^2 x \) is \(-\cot x\). Thus, for the first term: \[ \frac{1}{2} \int \csc^2 x \, dx = -\frac{1}{2} \cot x + C_1 \] where \( C_1 \) is a constant of integration.
03

Find Antiderivative for the Second Term

The antiderivative of \( \csc x \cot x \) is \(-\csc x\). Thus, for the second term: \[ -\frac{1}{2} \int \csc x \cot x \, dx = \frac{1}{2} \csc x + C_2 \] where \( C_2 \) is another constant of integration.
04

Combine Results

Combine the results from Steps 2 and 3. The most general antiderivative is:\[ -\frac{1}{2} \cot x + \frac{1}{2} \csc x + C \] where \( C = C_1 + C_2 \) is a general constant.
05

Check by Differentiation

Differentiate the result from Step 4: \[ \frac{d}{dx}\left(-\frac{1}{2}\cot x + \frac{1}{2}\csc x + C\right) = \frac{1}{2} \csc^2 x - \frac{1}{2} \csc x \cot x \] which is the original integrand, confirming the antiderivative is correct.

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Key Concepts

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

Definite Integrals
Definite integrals represent the accumulation of quantities, such as areas under curves, over specified intervals. However, in the context of this exercise, we're focusing on a different concept, the *indefinite integral*, which relates to finding the most general form of an antiderivative. Remember, while a definite integral computes a number or total quantity, an indefinite integral results in a function鈥攖he antiderivative.
When solving indefinite integrals, the result includes a constant of integration, denoted usually as "C". This constant represents all possible vertical displacements of the antiderivative since differentiation of a constant yields zero.
- Indefinite integrals are about finding a general form rather than a specific numerical answer. - Don't forget the constant of integration! It's crucial to denote the full family of potential solutions.
Trigonometric Functions
Trigonometric functions such as sine, cosine, tangent, cosecant (csc), secant, and cotangent (cot) form the foundation of many calculus problems. In our exercise, we're particularly concerned with less common functions:
  • **Cosecant** (\(\csc x\)): the reciprocal of sine (\(\csc x = \frac{1}{\sin x}\))
  • **Cotangent** (\(\cot x\)): the reciprocal of tangent, or the quotient of cosine over sine (\(\cot x = \frac{\cos x}{\sin x}\))
These functions have specific derivatives:
  • The derivative of \(\csc^2 x\) is related to tangents and cosecants.
  • The derivative of \(\csc x \cot x\) is straightforward \(- \csc^2 x\), thus direct when reversed in solving integrals.
Understanding the derivatives helps find antiderivatives, which is central in tackling this exercise.
Integration Techniques
To solve the given integral, some handy integration techniques are at play. Splitting an integrand into separate parts is a common first step. Let's break it down:
  • **Splitting the Integrand:** This involves separating a complex expression into simpler integrals, which can be solved individually, then combined. In our solution, \(\frac{1}{2}(\csc^2 x - \csc x \cot x)\) becomes two separate integrals.
  • **Applying Known Antiderivatives:** It's crucial to know the antiderivatives of standard functions. For example, the antiderivative of \(\csc^2 x\) is known as \(-\cot x\). Recognizing these patterns allows quick solving.
  • **Combining Results:** Once individual parts are integrated, they are combined back together, always including the constant of integration to reflect all possible solutions.
These techniques simplify the solution process, ensuring accurate results verified by differentiation.

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Most popular questions from this chapter

Find the values of constants \(a, b,\) and \(c\) so that the graph of \(y=a x^{3}+b x^{2}+c x\) has a local maximum at \(x=3,\) local minimum at \(x=-1,\) and inflection point at \((1,11) .\)

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{4}{3} x-\tan x, \quad \frac{-\pi}{2}< x<\frac{\pi}{2}$$

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Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=5 x^{2 / 5}-2 x$$

Distance between two ships At noon, ship \(A\) was 12 nautical miles due north of ship \(B .\) Ship \(A\) was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yd) and continued to do so all day. Ship \(B\) was sailing east at 8 knots and continued to do so all day. \begin{equation} \begin{array}{l}{\text { a. Start counting time with } t=0 \text { at noon and express the dis- }} \\ \quad {\text { tance } s \text { between the ships as a function of } t \text { . }} \\ {\text { b. How rapidly was the distance between the ships changing at }} \\ \quad {\text { noon? One hour later? }} \\\ {\text { c. The visibility that day was } 5 \text { nautical miles. Did the ships }} \\ \quad {\text { ever sight each other? }} \\ {\text { d. Graph } s \text { and } d s / d t \text { together as functions of } t \text { for }-1 \leq t \leq 3} \\ \quad {\text { using different colors if possible. Compare the graphs and }} \\ \quad {\text { reconcile what you see with your answers in parts (b) }} \\ \quad {\text { and (c). }} \\ {\text { e. The graph of } d s / d t \text { looks as if it might have a horizontal }} \\\ \quad {\text { asymptote in the first quadrant. This in turn suggests that }} \\\ \quad {d s / d t \text { approaches a limiting value as } t \rightarrow \infty . \text { What is this }} \\ \quad {\text { value? What is its relation to the ships' individual speeds? }} \end{array} \end{equation}
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