Problem 38 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(37-54\). $$ \int_{-1}^{0} \frac{3 d x}{3 x-2} $$

Step-by-Step Solution

Answer

The value of the integral is \( \ln \left(\frac{2}{5}\right) \).

Step 1: Identify the Type of Integral

The integral is \( \int_{-1}^{0} \frac{3}{3x-2} \, dx \), which is a rational function. It can be solved using a simple substitution.

Step 2: Choose a Substitution

Let \( u = 3x - 2 \). Then, calculate \( du \) as follows: \( \frac{du}{dx} = 3 \), which translates to \( du = 3 \, dx \).

Step 3: Adjust Differential Components

We replace \( dx \) with \( \frac{1}{3}du \) because \( 3 \, dx = du \). Also, adjust the limits of integration for \( u \): - When \( x = -1 \), \( u = 3(-1) - 2 = -5 \)- When \( x = 0 \), \( u = 3(0) - 2 = -2 \).

Step 4: Substitute and Simplify the Integral

The integral becomes \( \int_{-5}^{-2} \frac{3}{u} \cdot \frac{1}{3} \, du \). Simplifying it we get: \( \int_{-5}^{-2} \frac{1}{u} \, du \).

Step 5: Integrate the Function

Integrate \( \int \frac{1}{u} \, du \) to get \( \ln |u| \). Thus, the integral becomes \( \left[ \ln |u| \right]_{-5}^{-2} \).

Step 6: Evaluate the Definite Integral

Evaluate \( \left[ \ln |u| \right]_{-5}^{-2} \):- Substitute the upper limit: \( \ln |-2| = \ln 2 \)- Substitute the lower limit: \( \ln |-5| = \ln 5 \)- Thus, \( \ln 2 - \ln 5 = \ln \left(\frac{2}{5}\right) \).

Key Concepts

Substitution Method

The substitution method is a powerful technique for evaluating integrals, especially those involving rational functions, which can be complex. This method simplifies the integration by changing the variable of integration, using a substitution that transforms the original integral into a more manageable form. In the given problem, we identified the integral \( \int \frac{3}{3x-2} \, dx \) as a rational function that can be solved using substitution.The key steps in this method are:

• Choose a new variable, \( u \), to replace an expression in the integrand. This substitution should simplify the integrand.
• Compute \( du \) in terms of the original variable, adjusting the differential components appropriately.
• Transform the original integration limits to match the new variable \( u \).

By substituting \( u = 3x - 2 \), the differential becomes \( du = 3 \, dx \), allowing the transformation of the integral expression into a simpler form. This change often turns the integral into a standard form that is easier to integrate. The substitution method streamlines many complex integrals by simplifying them through strategic variable change.

Rational Function

Rational functions are expressed as the ratio of two polynomials, and they often appear in integration problems. Understanding how to manage their features is crucial for applying techniques such as substitution and partial fraction decomposition.In our original problem, the rational function \( \frac{3}{3x-2} \) needed manipulation to integrate effectively.Key characteristics of rational functions include:

• The numerator and denominator are both polynomials.
• They can contain asymptotes and discontinuities, which can affect the integration process.
• Integrating them often involves simplification methods like substitution or decomposition.

The function \( \frac{3}{3x-2} \) simplifies by using substitution which reduces the downstream integral into \( \int \frac{1}{u} \, du \), reflecting a standard form that is straightforward to solve.Recognizing the structure of rational functions allows for strategic transformations, such as the substitution used here, to make the integration process manageable.Mastering these transformations is essential when solving integrals involving rational functions.

Integration Limits

When dealing with definite integrals, integration limits are the boundaries over which you evaluate the integral. These limits are not just numbers but critical components that define the exact part of the function being integrated. They are usually represented as upper and lower bounds.In the original exercise, the limits were from \( x = -1 \) to \( x = 0 \).After substitution, these needed to be adjusted to the new variable \( u \) to ensure correct evaluation:

• For \( x = -1 \), \( u = 3(-1) - 2 = -5 \).
• For \( x = 0 \), \( u = 3(0) - 2 = -2 \).

Changing the limits of integration is as important as altering the integrand itself because they preserve the original calculation's meaning after substitution.Once these new limits are set, they guide the evaluation of the integral with respect to the new variable, ensuring an accurate and complete integration.Understanding how to adjust and apply integration limits after a substitution is a fundamental skill when solving definite integrals.