Problem 41 - Calculus Single Variable

Question

Calculate the given integral. \(\int \frac{2 x^{2}+4 x+9}{x^{3}-1} d x\)

Step-by-Step Solution

Answer

\( \int \frac{2x^2+4x+9}{x^3-1}dx = \ln|(x-1)(x^2+x+1)| + C \)

Step 1: Identify the Method

To solve the integral \( \int \frac{2x^2 + 4x + 9}{x^3 - 1} \, dx \), we first notice that the degree of the numerator is less than the denominator. Hence, we should use partial fraction decomposition to simplify the integrand.

Step 2: Factor the Denominator

The denominator can be factored as follows: \( x^3 - 1 = (x - 1)(x^2 + x + 1) \). With these factors, we can proceed to set up the partial fraction decomposition.

Step 3: Set Up Partial Fractions

We write the integrand as a sum of fractions: \( \frac{2x^2 + 4x + 9}{x^3 - 1} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + x + 1} \). Our goal is to solve for the constants \( A, B, \) and \( C \).

Step 4: Solve for Constants

Multiply both sides by the common denominator \( x^3 - 1 \) to clear the fractions, resulting in \( 2x^2 + 4x + 9 = A(x^2 + x + 1) + (Bx + C)(x - 1) \). Expanding and equating coefficients, we solve for \( A = 1, B = 2, C = 1 \).

Step 5: Integrate Separate Terms

The integral splits into two simpler integrals: \( \int \frac{1}{x-1}dx + \int \frac{2x + 1}{x^2 + x + 1}dx \). The first integral is \( \ln|x-1| + C_1 \). The second requires substitution or recognizing it as the derivative form: it evaluates to \( \ln(x^2 + x + 1) + C_2 \).

Step 6: Combine Results

Combine the integrated terms: \( \ln|x-1| + \ln(x^2 + x + 1) + C \). This can be expressed simply as \( \ln| (x - 1)(x^2 + x + 1) | + C \) where \( C \) is a constant of integration.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used in integral calculus to simplify complex rational expressions. It helps us break down a fraction into simpler, more manageable parts. This method is essential when the degree of the numerator is less than the degree of the denominator.
In this particular exercise, the denominator is factored as \(x^3 - 1 = (x - 1)(x^2 + x + 1)\), indicating that we have two main components to work with. To write the fraction \(\frac{2x^2 + 4x + 9}{x^3 - 1}\) as a sum of simpler fractions, partial fraction decomposition allows us to express this as:

• A fraction \(\frac{A}{x - 1}\), where \(A\) is a constant.
• Another fraction \(\frac{Bx + C}{x^2 + x + 1}\), with \(B\) and \(C\) as constants.

By clearing fractions using the original polynomial denominator and equating coefficients, we solve for \(A, B,\) and \(C\). This makes integration possible, as each fraction corresponds to simpler expressions that are much easier to integrate.

Integration Techniques

Integration techniques are strategies that help us find the integral of complex functions. Among these, partial fraction decomposition is just one. Other techniques may involve substitution, integration by parts, or recognizing standard integral forms.
In this exercise, after decomposing the expression, we face two simpler integrals:

• \(\int \frac{1}{x-1} \, dx\)
• \(\int \frac{2x + 1}{x^2 + x + 1} \, dx\)

For the first integral, we use the basic technique of recognizing the form \(\frac{1}{u} \, du = \ln|u| + C\), which directly integrates to a natural logarithm expression. For the second, substitution methods often help. However, in this problem, recognizing the derivative form greatly speeds up the process. If you suspect the numerator to be similar to the derivative of the denominator, it is often the best way to go.

Logarithmic Integration

Logarithmic integration refers to integrating functions that result in a natural logarithm. It's a standard outcome in calculus, especially when dealing with rational functions.
The first integral \(\int \frac{1}{x-1} \, dx\) straightforwardly results in \(\ln|x-1| + C_1\). Natural logarithms appear because when the derivative of the inside equals the numerator, the integral simplifies greatly to this logarithmic form.
For the second component \(\int \frac{2x + 1}{x^2 + x + 1} \, dx\), recognizing it as a derivative form results in \(\ln(x^2 + x + 1) + C_2\). This is a more complex form of logarithmic integration, but by recognizing the numerator's resemblance to the denominator's derivative, the integration aligns smoothly.
Combining these two results, the original integral \(\int \frac{2x^2 + 4x + 9}{x^3 - 1} \, dx\) simplifies to a natural log expression, summarizing the links between partial fractions, integration techniques, and logarithmic integration methods.