Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$ \int \frac{d x}{x \ln x}(x>1) $$

Short answer

\( \ln(\ln x) + C \)

Step-by-step solution

Identify the Substitution

Observe that the expression in the denominator contains both \( x \) and \( \ln x \). This suggests substituting the function inside the logarithm. Let \( u = \ln x \). Then, the derivative \( du = \frac{1}{x} \, dx \).

Rewrite the Integral using Substitution

With the substitution \( u = \ln x \), we have \( du = \frac{1}{x} \, dx \), implying \( dx = x \, du \). Therefore, \( x = e^u \) and \( dx = e^u \, du \). Substitute these into the integral: \[ \int \frac{1}{e^u u} e^u du = \int \frac{du}{u} \].

Integrate the Simplified Integral

The integral \( \int \frac{du}{u} \) is a standard integral, and its solution is \( \ln |u| + C \), where \( C \) is the constant of integration.

Resubstitute the Original Variable

Since we substituted \( u = \ln x \), replace \( u \) back with \( \ln x \). This gives the expression \( \ln |\ln x| + C \). Since \( x > 1 \), \( \ln x \) is positive, allowing us to drop the absolute value, resulting in \( \ln(\ln x) + C \).

Key concepts

Integration by Substitution

Integration by substitution is a powerful technique used to simplify an integral by making a change of variable. The aim is to rewrite a complex integral into a simpler form that is easier to evaluate. This method hinges on the idea of reversing the chain rule for derivatives.

To use this technique, identify a portion of the integral that can be replaced with a new variable, usually noted as "u". In our exercise, we saw the choice of substituting the natural logarithm: let \( u = \ln x \).

• Differentiate \( u \) to find \( du \). For our example, \( du = \frac{1}{x} \, dx \).
• Rewrite the integral in terms of \( u \) and \( du \). This helps convert the integral into a simpler form.

The purpose of substitution is to transform the given function, making it easier to work with and solve, like how our integral transformed into \( \int \frac{du}{u} \). The key is to carefully undo the substitution after integrating, which allows us to write the solution in terms of the original variable.

Logarithmic Integration

Logarithmic integration involves integrating functions that include logarithmic components. The natural logarithm, denoted by \( \ln \), is commonly involved in these types of integrals.

In our exercise, we derived the integral \( \int \frac{du}{u} \). This is a standard logarithmic integral, whose solution is \( \ln |u| + C \), where \( C \) represents the constant of integration.

Logarithmic integration is closely related to substitution because often, simplifying an integral into a logarithmic form involves substitution. In many calculus exercises, recognizing when a natural logarithm is the antiderivative is crucial for solving the problem.

• Remember that using logarithmic rules can simplify integrals containing logarithmic functions.
• Ensure you apply properties of logarithms when integrating, such as recognizing \( \frac{d}{dx} (\ln x) = \frac{1}{x} \).

Converting a challenging integral into a basic form like \( \int \frac{du}{u} \) highlights the elegance of logarithmic integration.

Calculus Exercises

Calculus exercises are designed to strengthen your understanding and application of calculus concepts through practice. They often incorporate a variety of techniques like substitution and integration.

Working through exercises such as \( \int \frac{dx}{x \ln x} \), you'll improve your skills in several areas:

• Identifying appropriate substitutions for simplifying integrals.
• Understanding how to maneuver between different forms of integration, such as polynomial and logarithmic.
• Gaining practice in integrating complex expressions and finding antiderivatives.

In calculus, mastering the exercises is crucial because it helps you recognize patterns and the best strategies to tackle similar problems in the future.

Engaging with a variety of problems will build confidence, allowing you to handle more complicated calculus tasks effectively.