Problem 16 - Calculus Single Variable

Question

Use the method of substitution to calculate the indefinite integrals. $$ \int \frac{\ln (x)}{x} d x $$

Step-by-Step Solution

Answer

\( \int \frac{\ln(x)}{x} \, dx = \frac{(\ln(x))^2}{2} + C \)

Step 1: Choose a substitution

For substitution, let's set \[ u = \ln(x) \]This choice will simplify the integral because the derivative of \( u \) with respect to \( x \), \( du \), should help cancel parts of the integrand.

Step 2: Differentiate the substitution

Differentiate \( u = \ln(x) \) with respect to \( x \):\[ \frac{du}{dx} = \frac{1}{x} \]Rewriting, we have \( du = \frac{1}{x} \, dx \). This implies that \( dx = x \, du \).

Step 3: Substitute in the integral

Substitute \( u = \ln(x) \) and \( dx = x \, du \) into the original integral:\[ \int \frac{\ln(x)}{x} \, dx = \int u \cdot \frac{1}{x} \cdot x \, du = \int u \, du \]

Step 4: Integrate with respect to \( u \)

Now, integrate the function with respect to \( u \):\[ \int u \, du = \frac{u^2}{2} + C \]Here, \( C \) is the constant of integration.

Step 5: Substitute back to \( x \)

Replace \( u \) with \( \ln(x) \) to express the solution in terms of the original variable:\[ \frac{u^2}{2} + C = \frac{(\ln(x))^2}{2} + C \]

Step 6: Write the final answer

Thus, the indefinite integral evaluates to:\[ \int \frac{\ln(x)}{x} \, dx = \frac{(\ln(x))^2}{2} + C \]

Key Concepts

Method of Substitution

The method of substitution is a powerful technique for solving integrals that might initially seem complex. It involves replacing a part of the integrand with a new variable, often denoted as \( u \), in order to simplify the integration process. This method is especially useful when dealing with composite functions or when one part of the integrand has a derivative elsewhere in the equation.

In the exercise, the substitution \( u = \ln(x) \) is chosen. This choice is strategic because the derivative of \( \ln(x) \), which is \( \frac{1}{x} \), appears in the integrand as \( \frac{1}{x} \, dx \). Hence, making the substitution simplifies the integral significantly because the integration becomes straightforward with respect to \( u \).

The key steps while using this method include:

• Pick a substitution that simplifies the integral.
• Differentiate that substitution to find \( du \).
• Rewrite the integral in terms of \( u \).
• Integrate with respect to \( u \).
• Substitute back the original variable.

Logarithmic Integration

Logarithmic integration refers to the process of integrating functions that involve logarithms, particularly the natural logarithm \( \ln(x) \). Logarithmic terms often appear in calculus, and recognizing them can help decide whether substitution is an appropriate approach.

For the given exercise, \( \int \frac{\ln(x)}{x} \, dx \) we specifically have a logarithm \( \ln(x) \) divided by \( x \), which is a classic scenario for substitution where the method leads to an integrand \( \int u \, du \) that is manageable. This approach demonstrates how logarithmic functions can transform into polynomial forms that are easier to integrate.

When integrating logarithmic functions, identifying parts of the integral that match the derivative of \( \ln(x) \), like \( \frac{1}{x} \), is crucial. This makes it straightforward to apply substitution and integrate easily.

Remember, after finding the indefinite integral, always verify your solution by differentiating it. This step ensures that you have correctly executed the integration.

Change of Variables

Change of variables is a core concept related to the method of substitution, and it involves altering the variable of integration to simplify the integral. This technique is especially helpful for functions where parts of the expression can be associated with known derivatives.

In this context, changing variables from \( x \) to \( u = \ln(x) \) transformed our integrand into a much simpler form. By recognizing that \( du = \frac{1}{x} \, dx \), we could easily rewrite the original integral \( \int \frac{\ln(x)}{x} \, dx \) in terms of \( u \). This reduced the problem to a simple polynomial integral \( \int u \, du \).

The main steps in changing variables include:

• Identifying a substitution that makes the integrand easier to handle.
• Expressing \( dx \) in terms of \( du \) and the new variable.
• Rewriting the integral in terms of this new variable.

This technique can be applied to a wide variety of integrals, making it a versatile tool in calculus.

Integration Techniques

Integration techniques, such as substitution and change of variables, are essential skills in calculus used to solve a vast array of integral problems. These techniques help transform difficult integrals into simpler forms that can be easily evaluated.

The integration of \( \int \frac{\ln(x)}{x} \, dx \) illustrates how a seemingly complex problem can be reduced to a basic polynomial integration using strategic substitutions. This highlights how various techniques can apply based on the function's structure.

Common integration techniques include:

• Substitution: Transforming the integral to a simpler form with a new variable.
• Integration by parts: Useful for integrals of products of functions.
• Partial fraction decomposition: Decomposes rational functions into simpler fractions for easier integration.
• Trigonometric identities: Simplify integrals involving trigonometric functions.

Mastering these techniques involves practice and recognition of patterns within the integrand that suggests the best approach. Utilizing these methods efficiently can simplify complex calculus problems and broaden understanding within mathematics.