Problem 32 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{(\ln z)^{2}}{z} d z $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integration Technique

The integral \( \int \frac{(\ln z)^{2}}{z} \, dz \) can be solved using the technique of substitution. Recognize that if we set \( u = \ln z \), then \( du = \frac{1}{z} \, dz \). This substitution simplifies the integral.

Step 2: Substitute and Simplify the Integral

Replace \( \ln z \) with \( u \) and \( \frac{1}{z} \, dz \) with \( du \). The integral becomes \( \int u^2 \, du \).

Step 3: Integrate with Respect to u

Integrate \( \int u^2 \, du \) using the power rule. The power rule states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). For \( u^2 \), this gives \( \frac{u^3}{3} + C \).

Step 4: Substitute Back to Original Variable

Substitute back \( u = \ln z \) into the integrated result, giving \( \frac{(\ln z)^3}{3} + C \).

Step 5: Differentiate to Check the Result

Differentiate the result \( \frac{(\ln z)^3}{3} + C \) with respect to \( z \). Using the chain rule, \( \frac{d}{dz} \left( \frac{(\ln z)^3}{3} \right) = (\ln z)^2 \cdot \frac{1}{z} \), which matches the original integrand.

Key Concepts

Substitution Method

The substitution method is a powerful integration technique used to simplify complex integrals by introducing a new variable. In the case of the integral \( \int \frac{(\ln z)^{2}}{z} \, dz \), substitution is particularly useful. The trick is to identify a part of the integrand that, when substituted with a new variable, simplifies the integral. Here's a step-by-step of how this works:

• Identify a part of the function that, when substituted, reduces complexity. For example, if \( u = \ln z \).
• Calculate the differential of the substitute variable, which gives us \( du = \frac{1}{z} \, dz \).
• Transform the entire integrand into terms of \( u \) and \( du \). So, the integral becomes \( \int u^2 \, du \).

By using substitution, we reduce the problem into a simpler form that is easier to integrate.

Power Rule

Once the integral is transformed using substitution, you often need to solve it using the power rule for integration. The power rule is a basic principle that applies to polynomials and is crucial in calculus. It states that the integral of \( u^n \) with respect to \( u \) is given by:\[\int u^n \, du = \frac{u^{n+1}}{n+1} + C\]In our example, after substitution, the integral becomes \( \int u^2 \, du \). Applying the power rule, we integrate:

• Raise the power of \( u \) by one, resulting in \( u^3 \).
• Divide by the new power, which gives us \( \frac{u^3}{3} \).
• Add the constant of integration \( C \).

This simplification underscores the elegance of the power rule, allowing for the integration of polynomial expressions with ease.

Differentiation Check

After integrating a function, it is a good practice to verify the correctness of the result by differentiating it back to the original function. This process ensures that no errors were made during the integration process. Here's how the differentiation check works for \( \frac{(\ln z)^3}{3} + C \):

• Differentiate by applying the chain rule: for \( \ln z \) raised to a power, first bring down the exponent \( 3 \) and multiply by the derivative of \( \ln z = \frac{1}{z} \).
• This yields: \( 3 \times \frac{1}{3} \times (\ln z)^2 \times \frac{1}{z} = (\ln z)^2 \times \frac{1}{z} \).
• The computed derivative \( (\ln z)^2 \times \frac{1}{z} \) confirms it is indeed identical to the original integrand.

The differentiation check reassures that your solution is correct and acts as a valuable tool for validating integration problems.