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Chapter 4: Problem 33

Evaluate. Assume \(u>0\) when In \(u\) appears. (Be sure to check by differentiating!) $$\int \frac{d x}{x \ln x}$$

Short Answer

Expert verified
The integral evaluates to .

Step by step solution

01

- Identify the integral

We start with the integral .
02

- Apply substitution

Let us use the substitution method. Define the substitution as follows: .This means, we need to find .
03

- Substitute and simplify

Now substitute and simplify: .
04

- Evaluate the integral

Evaluate the integral: = .
05

- Back-substitute and simplify

Substitute back . This gives us: .
06

- Add constant of integration

Finally, we add the constant of integration , giving us:
07

- Differentiate to verify

Differentiate with respect to . Correct differentiation should give us the original integrand, verifying our solution.

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique in calculus for simplifying integrals. The idea is to change a complicated integral into a simpler one by substituting part of the original integrand with a new variable. In our exercise, we dealt with the integral \( \int \frac{dx}{x \ln x} \).
To use substitution:
  • Choose a substitution that makes the problem simpler. In this case, we let \( u = \ln x \).
  • Determine the derivative of your substitution. If \( u = \ln x \), then \( du = \frac{1}{x} dx \).
  • Rewrite the integral in terms of the new variable \( u \). For this integral, \( \frac{dx}{x \ln x} \) becomes \( \int \frac{du}{u} \).
  • Integrate the simpler integral. The integral \( \int \frac{du}{u} \) is straightforward and equals \( \ln |u| + C \).
  • Finally, substitute back the original variable to obtain the answer: \( \ln |\ln x| + C \).
This technique helps break down complex integrals, making them more manageable.
Natural Logarithm
The natural logarithm, denoted as \( \ln x \), is a logarithm to the base \( e \) (where \( e \) is approximately 2.71828).
Some key properties of the natural logarithm are:
  • \( \ln 1 = 0 \) because \( e^0 = 1 \).
  • \( \ln(e^x) = x \) for any real number \( x \). This is due to the definition of \( e \) as the base of the natural logarithm.
  • \( \frac{d}{dx} \ln x = \frac{1}{x} \). This property of differentiation is crucial in integral calculations, as seen in our exercise.
In the context of our exercise, substituting \( u = \ln x \) simplified the integral. The natural logarithm helps transform a product into a sum (or a quotient into a difference), which is often simpler to integrate.
Constant of Integration
When performing indefinite integration, you must add a constant of integration, denoted by \( C \). The reason for this is that integration is the reverse process of differentiation, and differentiating a constant term gives zero.
Consider the integral \( \int \frac{du}{u} \). The most general form of its antiderivative is \( \ ln |u| + C \), where \( C \) is any constant.
  • In our problem, the constant of integration is added after integrating \( \int \frac{du}{u} \).
  • We write the final answer as \( \ln |\ln x| + C \) after substituting back \( u = \ln x \).
This constant is essential in representing all possible antiderivatives of a function. It ensures that we capture every function that could have the given integrand as its derivative.

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Most popular questions from this chapter

Evaluate. \(\int_{0}^{2} \sqrt{2 x} d x \quad(\text { Hint: Simplify first. })\)

Find \(s(t)\) $$a(t)=-2 t+6, \text { with } v(0)=6 \text { and } s(0)=10$$

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Find \(s(t)\) $$v(t)=2 t, \quad s(0)=10$$
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