Problem 32 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{(\ln x)^{2}}{x} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{(\ln x)^3}{3} + C \).

Step 1: Recognize the Integral Form

The integral given is \( \int \frac{(\ln x)^2}{x} \, dx \), which can be recognized as a candidate for integration by substitution because the derivative of \( \ln x \) appears in the integrand.

Step 2: Choose a Substitution

Set \( u = \ln x \), so that \( du = \frac{1}{x} \, dx \). This substitution transforms the original integral.

Step 3: Transform the Integral

When we substitute \( u = \ln x \), the integral becomes \( \int u^2 \, du \). This is now a straightforward polynomial integral.

Step 4: Integrate

The integral of \( u^2 \) is \( \frac{u^3}{3} + C \). Thus, replacing back \( u = \ln x \), we get \( \frac{(\ln x)^3}{3} + C \).

Step 5: Differentiate to Check the Result

Differentiate \( \frac{(\ln x)^3}{3} + C \) with respect to \( x \). Using the chain rule, the derivative is \( \frac{1}{3} \cdot 3 (\ln x)^2 \cdot \frac{1}{x} = \frac{(\ln x)^2}{x} \), which matches the original integrand.

Key Concepts

Polynomial Integral

Understanding polynomial integrals is essential for working through calculus exercises like the one provided. A polynomial is essentially an algebraic expression made up of variables raised to the power of whole numbers and can include constants. In this exercise, by substituting, we transformed the integral \[ \int \frac{(\ln x)^{2}}{x} \, dx \]into a much simpler form, \( \int u^2 \, du \), where \( u = \ln x \).

Polynomial integrals often follow a straightforward rule:

• To integrate \( u^n \), use \( \frac{u^{n+1}}{n+1} + C \)

Here, \( u^n \) was \( u^2 \), so integrating it gave us \( \frac{u^3}{3} + C \). Replacing \( u \) with \( \ln x \), we revert the expression back to terms of \( x \) to complete our solution. Converting to a polynomial integral greatly simplifies the integration process and provides a clearer pathway to the solution.

Chain Rule

The chain rule in calculus is a powerful tool used to differentiate composite functions. It allows us to handle functions embedded within other functions by differentiating the outer function and then multiplying by the derivative of the inner function.

In this exercise, after integrating, we differentiated the result to check its correctness. Our final integral result was:\[ \frac{(\ln x)^3}{3} + C \]To differentiate this expression with respect to \( x \), we applied the chain rule.

• First, differentiate the outer function, \( \frac{u^3}{3} \), giving us \( \frac{1}{3} \, 3u^2 \)
• Then, multiply by the derivative of the inner function, \( u = \ln x \), which is \( \frac{1}{x} \)

This operation returned us to the original integrand. Therefore, the chain rule here ensured that the integration process was done correctly, confirming our solution's accuracy.

Differentiation Check

Checking your work is a crucial part of solving calculus problems. In integration, we often utilize differentiation to verify our solution. This means differentiating the result of our integral to see if we return to the original integrand.

For this exercise, our differentiation check involved the expression:\[ \frac{(\ln x)^3}{3} + C \]When differentiated, each step should precisely reverse our initial integration. Here is what we did:

• Applying differentiation to \( \frac{(\ln x)^3}{3} \), gave us \( \frac{1}{3} \, 3(\ln x)^2 \, \frac{1}{x} \)
• This simplified back to \( \frac{(\ln x)^2}{x} \)

Thus, the differentiated result exactly matched the original integrand, proving our solution was correct. Differentiation checks are an important step in ensuring the integrity and accuracy of your calculus work.