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Chapter 4: Problem 39

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. $$\int\left(-3 \csc ^{2} x\right) d x$$

Short Answer

Expert verified
The antiderivative is \( 3 \cot x + C \).

Step by step solution

01

Identify the Integral Form

The integral given is \( \int (-3 \csc^2 x) \, dx \). Recognize that \( -3 \csc^2 x \) is similar to the derivative of a known trigonometric function.
02

Recall Derivative Identity

Recall the identity: the derivative of \( \cot x \) is \( -\csc^2 x \). Thus, \( -3 \csc^2 x \) is the derivative of \( 3 \cot x \).
03

Write the Antiderivative

Using the identified derivative, the antiderivative of \( -3 \csc^2 x \) is \( 3 \cot x + C \), where \( C \) is the constant of integration.
04

Check by Differentiation

Differentiate \( 3 \cot x + C \) with respect to \( x \). The derivative is \( 3(-\csc^2 x) = -3 \csc^2 x \), confirming the correctness of the antiderivative.

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Key Concepts

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

Indefinite Integrals
Indefinite integrals are a very important concept in calculus. They are essentially the reverse process of differentiation. When you find an indefinite integral, you determine the family of functions whose derivative is the given function.An indefinite integral is represented by the integral sign followed by a function and the differential (e.g., \( \int f(x) \, dx \)). This process is also called finding the antiderivative of the function. The result is a general antiderivative plus the constant of integration, \( C \). This constant is essential because differentiation removes constants, so when integrating, we need to account for any potential constant that might have been in the original function.Here are a few key points about indefinite integrals:
  • They represent a family of functions, all of which differ by a constant.
  • They can be used to solve problems involving accumulation, such as finding area or distance.
Trigonometric Functions
Trigonometric functions are functions of an angle and are fundamental in mathematics because they describe the relationships between the angles and sides of triangles.In calculus, we frequently encounter integrals involving trigonometric functions because they often appear in various mathematical and real-world problems. Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Functions like cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)) are reciprocals of these and also play significant roles.In the context of integration, recognizing the derivatives of these functions can quickly point to their antiderivatives. For example, the derivative of \( \cot x \) is \( -\csc^2 x \), which helps identify that the integral of \( -3 \csc^2 x \) becomes \( 3 \cot x + C \).Trigonometric functions are used in various fields:
  • Physics: For modeling waves and oscillations.
  • Engineering: To analyze periodic signals.
  • Architecture: For calculating angles in structures.
Differentiation
Differentiation is a key operation in calculus, used to find the rate at which a function is changing. When you differentiate a function, you are essentially finding the slope of the tangent line at any point on a graph of the function.The process of differentiation calculates the derivative, which is the main tool for finding the velocity in physics given a position function, or the rate of change of economic indicators.Checking our indefinite integral solutions often involves differentiating the result because differentiation reverses the process of integration. For example, in the exercise we reviewed,
  • The antiderivative found was \( 3 \cot x + C \).
  • Differentiating \( 3 \cot x \) gives us back \( -3 \csc^2 x \), confirming our integral calculation was correct.
Differentiation highlights:
  • It's used to find maximum and minimum values of functions.
  • Very useful in optimization problems and motion analysis.
  • A fundamental concept in calculus, allowing the understanding of change in various models and systems.

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Most popular questions from this chapter

The 800 -room Mega Motel chain is filled to capacity when the room charge is \(\$ 50\) per night. For each \(\$ 10\) increase in room charge, 40 fewer rooms are filled each night. What charge per room will result in the maximum revenue per night?

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Let \(f\) be differentiable at every value of \(x\) and suppose that \(f(1)=1,\) that \(f^{\prime} < 0\) on \((-\infty, 1),\) and that \(f^{\prime} > 0\) on \((1, \infty)\) . a. Show that \(f(x) \geq 1\) for all \(x\) b. Must \(f^{\prime}(1)=0 ?\) Explain.

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