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Chapter 6: Problem 26

In Exercises 1 through 38 , find the antiderivatives. $$ \int \frac{2}{\pi x} d x $$

Short Answer

Expert verified
The antiderivative is \( \frac{2}{\pi} \ln|x| + C \).

Step by step solution

01

Identify the Integral Form

The given integral is \( \int \frac{2}{\pi x} \, dx \). This can be viewed as \( \frac{2}{\pi} \int \frac{1}{x} \, dx \) because \( \frac{2}{\pi} \) is a constant that can be factored out.
02

Recognize the Basic Antiderivative Formula

Recall that the antiderivative of \( \frac{1}{x} \, dx \) is \( \ln|x| + C \), where \( C \) is the constant of integration.
03

Apply the Scale Factor to the Antiderivative

Apply the constant factor \( \frac{2}{\pi} \) to the basic antiderivative. Therefore, the antiderivative of \( \int \frac{2}{\pi x} \, dx \) is \( \frac{2}{\pi} \ln|x| + C \), where \( C \) is the integration constant.

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Key Concepts

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

Integration
Integration is a fundamental concept in calculus. It is the process of finding antiderivatives or integrals. The integral can be thought of as the opposite of the derivative, and it is used to find the area under a curve.

In the given exercise, we are tasked with finding the antiderivative of the expression \( \int \frac{2}{\pi x} \, dx \). Here, the integration process involves understanding how constants and functions behave during integration. Notice how the constant \( \frac{2}{\pi} \) is factored out to simplify the function inside the integral.
  • Integration simplifies expressions by handling constant terms separately.
  • Integrals like \( \int \frac{1}{x} \, dx \) are key to understanding logarithmic functions.
  • Antiderivatives always include an integration constant \( C \) because differentiation nullifies constants.
Natural Logarithm
The natural logarithm function, written as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is approximately 2.71828. It plays a crucial role in various mathematical calculations, especially in calculus.

In this exercise, recognizing that the integral of \( \frac{1}{x} \, dx \) is \( \ln|x| \) is essential for solving problems involving logarithmic functions. The natural logarithm helps in reversing the differentiation of exponential functions. Thus, when you integrate a function involving \( \frac{1}{x} \), the result is connected to the natural logarithm.
  • Natural logarithms are used to simplify integration involving fraction functions.
  • \( \ln|x| \) is derived from the absence of any base other than \( e \), making it unique.
  • The absolute value symbols \( |x| \) ensure the logarithm is defined for all possible values of \( x \).
Calculus
Calculus is the branch of mathematics that studies change. It encompasses both differentiation and integration. Differentiation measures rates of change, while integration accumulates quantities.

In this particular problem, we applied calculus techniques to identify and apply the antiderivative of \( \int \frac{2}{\pi x} \, dx \). The problem involves the direct application of basic integration rules, such as factoring out constants and identifying standard integral forms that yield logarithmic functions.
  • Calculus provides tools for solving practical problems in physics, engineering, and economics.
  • Understanding both integration and differentiation is crucial for mastering calculus.
  • Problems like this one demonstrate the elegance and power of calculus in simplifying complex mathematics.

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Most popular questions from this chapter

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