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Chapter 5: Problem 43

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. \(\int \frac{\ln x}{x} d x \quad[\) Hint \(:\) Let \(u=\ln x .]\)

Short Answer

Expert verified
The integral is \( \frac{(\ln{x})^2}{2} + C \).

Step by step solution

01

Identify the Substitution

Given the hint, we choose the substitution \( u = \ln{x} \). This implies that the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = \frac{1}{x} \).
02

Rearrange for Differential

From \( \frac{du}{dx} = \frac{1}{x} \), we rearrange for \( dx \) to obtain \( du = \frac{1}{x} dx \) or \( dx = x \, du \).
03

Substitute in the Integral

Substitute \( u = \ln{x} \) and \( dx = x \cdot du \) back into the integral. Since \( \frac{dx}{x} = du \), the integral becomes \( \int u \, du \).
04

Integrate with Respect to u

The integral \( \int u \, du \) is a standard power rule integral. The antiderivative is \( \frac{u^2}{2} + C \), where \( C \) is the constant of integration.
05

Substitute Back to Original Variable

Replace \( u \) with \( \ln{x} \) to return to the original variable: \( \frac{(\ln{x})^2}{2} + C \). Thus, the indefinite integral is \( \frac{(\ln{x})^2}{2} + C \).

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Key Concepts

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

Substitution Method in Integration
The substitution method is a powerful technique used in integral calculus to simplify finding antiderivatives. When we have a complex integral, sometimes substituting a part of the function with a new variable makes the integration process more manageable. In our exercise, we use the hint to let \( u = \ln x \). This simplifies the integral and allows us to deal with a less complicated expression.
  • Step 1 involves choosing a substitution that captures the complexity of the integral. Here, the derivative of \( \ln x \) is simple, \( \frac{1}{x} \), making it a prime candidate for substitution.

  • After substitution, the integral's variable is changed from \( x \) to \( u \), transforming the integral into a form that's easier to solve.
This method is especially useful when dealing with integrals where direct methods are too cumbersome or difficult to apply directly.
Understanding Derivatives
A derivative represents the rate of change of a function. In the context of integration, derivatives help us choose appropriate substitutions and solve for differentials. In our example, noticing that the derivative of \( \ln x \) is \( \frac{1}{x} \) is crucial.
  • This insight allows us to express \( dx \) in terms of \( du \) as \( dx = x \, du \).

  • This step is vital because it transforms the integrand into a function of \( u \), enabling simpler integration.
Thus, understanding how derivatives relate to the function you're integrating helps in successfully applying the substitution method.
Antiderivative: Reversing Derivatives
An antiderivative is essentially the opposite of a derivative. It is a function whose derivative gives back the function from which we started. Finding an antiderivative is the critical step in determining the indefinite integral.In our exercise, after substituting \( u = \ln x \), we are left with the integral \( \int u \, du \). This involves using the power rule for integration: when you integrate \( u^n \), you increase the power by one and divide by the new power.
  • The antiderivative of \( u \) is calculated as \( \frac{u^2}{2} + C \), where \( C \) is the constant of integration.

  • This formulaic step is integral to finding the final answer by ensuring your integrated function differentiates back to the original form.
Integral Calculus: The Indefinite Integral
Integral calculus is the branch of mathematics dealing with integrals and their applications. The indefinite integral, unlike the definite integral which calculates an area, results in a general formula for the antiderivative of a function. In our exercise, we started with \( \int \frac{\ln x}{x} \, dx \) which we solved using a substitution technique. After finding the antiderivative in terms of \( u \), we substitute back the original variable to get the indefinite integral: \( \frac{(\ln x)^2}{2} + C \).
  • This indefinite integral expression represents all possible antiderivatives of the given function.

  • The "+ C" is significant because it accounts for any constant that could have been part of the original function but disappears upon differentiation.
Understanding integral calculus, and how to navigate through substitution and anti-differentiation, builds a solid foundation in solving complex mathematical problems.

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