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Chapter 5: Problem 22

Find each indefinite integral. \(\int\left(e^{2 x}-\frac{2}{x}\right) d x\)

Short Answer

Expert verified
\(\int\left(e^{2x} - \frac{2}{x}\right) dx = \frac{1}{2}e^{2x} - 2\ln|x| + C\) where \(C\) is the constant of integration.

Step by step solution

01

Break Down the Integral

The integral can be broken down into two separate integrals: \[\int\left(e^{2x} - \frac{2}{x}\right) dx = \int e^{2x} \, dx - \int \frac{2}{x} \, dx\]This allows us to handle each part individually.
02

Solve the First Integral

The first integral is \(\int e^{2x} \, dx\). To solve this, we use substitution. Let \(u = 2x\), then \(du = 2 \, dx\) or \(dx = \frac{du}{2}\):\[\int e^{2x} \, dx = \int e^u \cdot \frac{1}{2} \, du = \frac{1}{2} \int e^u \, du = \frac{1}{2} e^u + C_1\]Substituting back \(u = 2x\), we get:\[\frac{1}{2} e^{2x} + C_1\]
03

Solve the Second Integral

The second integral is \(\int \frac{2}{x} \, dx\). This is a standard integral that evaluates to:\[2 \cdot \int \frac{1}{x} \, dx = 2 \ln|x| + C_2\]
04

Combine the Results

Now that we have solved both integrals, combine them to find the indefinite integral:\[\int\left(e^{2x} - \frac{2}{x}\right) dx = \frac{1}{2}e^{2x} - 2\ln|x| + C\]Here, \(C\) represents the constant of integration, combining both \(C_1\) and \(C_2\).

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Key Concepts

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

Calculus
Calculus is a branch of mathematics that studies continuous change.
It helps us understand many physical phenomena and has applications in science, engineering, and everyday problem-solving. Indefinite integrals are an essential part of calculus.
They help find a family of functions whose derivative gives the integrand.
  • An indefinite integral is denoted by the integral sign \( \int \), followed by a function and the differential \( dx \).
  • The result, expressed as a function, represents possible solutions related by a constant \( C \), known as the constant of integration.
In the exercise above, you will learn how to handle a complex expression by splitting it into simpler components.
This technique allows for easier computation by separately solving each part of the integral.
Integration by Substitution
Integration by substitution is a technique used to simplify an integration problem. When faced with a difficult integral, we can use substitution to transform it into a simpler one.
It's similar to the reverse process of the chain rule in differentiation.When performing integration by substitution:
  • We choose a substitution \( u = g(x) \) that simplifies the integral.
  • The differential \( du \) is expressed in terms of \( dx \). This step is crucial as it helps in changing the variable for easier integration.
In the step-by-step solution, the substitution \( u = 2x \) helps transform the integral \( \int e^{2x} \, dx \) into a simpler integral, \( \int e^u \, du \).
This substitution allows us to use the basic rule that the integral of \( e^u \) is itself, leading to a straightforward solution.
Logarithmic Integration
Logarithmic integration comes into play when dealing with integrals of the form \( \int \frac{1}{x} \, dx \). This type of integral results in a natural logarithm of \( x \).
Recognizing this form is key to solving such integrals quickly.When integrating functions that resemble \( \frac{2}{x} \):
  • We take the constant factor out, simplifying the expression to \( 2 \int \frac{1}{x} \, dx \).
  • This leads naturally to the formula \( 2 \ln|x| \), incorporating the absolute value to ensure the logarithm is valid for both positive and negative values of \( x \).
Combining this result with the computed integral from the substitution provides a complete solution to the problem. The constant \( C \) in the final answer accounts for all possible antiderivatives.

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Most popular questions from this chapter

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