Question

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{3 x^{2}+4 x-6}{x^{2}-3 x+2} d x$$

Short answer

Question: Evaluate the integral \(\int{\frac{3x^2+4x-6}{x^2-3x+2}} dx\). Answer: The integral evaluates to \(\int{\frac{3x^2+4x-6}{x^2-3x+2}} dx = 3x + 14\ln{|x-1|} - \ln{|x-2|} + C\).

Step-by-step solution

Perform long division

First, we perform long division to simplify the given rational function. Long division of \(\frac{3x^2+4x-6}{x^2-3x+2}\): ________________________ \((x^2-3x+2) | 3x^2+4x-6\) Subtract \(3(x^2-3x+2)\): ________________________ \((x^2-3x+2) | 3x^2+4x-6\) \( - (3x^2-9x+6)\) ________________________ \(13x-12\) So, the result of the long division is: \(3 + \frac{13x-12}{x^2-3x+2}\).

Factor the denominator

Next, we factor the polynomial in the denominator: \(x^2 - 3x + 2 = (x-1)(x-2)\)

Apply partial fraction decomposition

We can now express \(\frac{13x-12}{(x-1)(x-2)}\) as partial fractions: \(\frac{13x-12}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2}\) Clearing the denominators, we have: \(13x - 12 = A(x-2) + B(x-1)\) Now, we can find the constants \(A\) and \(B\). Let's plug in values of \(x\) that make one of the terms zero: If \(x=1\), the equation becomes: \(1 = A(1-2) + B(1-1)\) \(1 = -A\) If \(x=2\), the equation becomes: \(14 = A(2-1) + B(2-2)\) \(14 = A\) Thus, we have found \(A=14\) and \(B=-1\), so the partial fraction decomposition is: \(\frac{13x-12}{(x-1)(x-2)} = \frac{14}{x-1} - \frac{1}{x-2}\)

Integrate the simplified function

Now we integrate the simplified function: $$\int{\frac{3x^2+4x-6}{x^2-3x+2}} dx = \int{(3 + \frac{13x-12}{(x-1)(x-2)})} dx$$ Separate the integrand into two individual integrals: $$\int{3dx} + \int{(\frac{14}{x-1} - \frac{1}{x-2})dx}$$ Now, we can easily integrate both parts: $$3x + 14\int{\frac{1}{x-1}}dx - \int{\frac{1}{x-2}}dx$$ $$3x + 14\ln{|x-1|} - \ln{|x-2|} + C$$ The final solution is: $$\int{\frac{3x^2+4x-6}{x^2-3x+2}} dx = 3x + 14\ln{|x-1|} - \ln{|x-2|} + C$$

Key concepts

Long Division

Long division is a mathematical process that simplifies the division of polynomials by reducing one polynomial in the fraction to a smaller degree. This technique is similar to long division with numbers but focuses on coefficients of polynomials.

When dealing with rational functions, particularly those in which the degree of the numerator is equal to or larger than the degree of the denominator, performing long division helps break down the original function. Here, we apply long division to \(\frac{3x^2+4x-6}{x^2-3x+2}\).

• We divide \(3x^2+4x-6\) by \(x^2-3x+2\).
• Start by dividing the leading terms to get the first part of the quotient, \(3\).
• Multiply and subtract \(3(x^2-3x+2)\) from the original numerator.

The result is a simpler expression: \(3 + \frac{13x-12}{x^2-3x+2}\). Long division transforms complicated rational expressions into simpler forms that are easier to integrate.

Rational Functions

Rational functions are mathematical expressions of one polynomial divided by another. We often encounter them in calculus and algebra. Understanding how to manipulate and simplify these functions is crucial.

• Consists of a numerator and a denominator, both of which are polynomials.
• The degree of the polynomial is significant in determining the complexity and approach for simplification.

When the degree of the numerator is greater than or equal to the degree of the denominator, use long division to simplify. In certain cases, factoring the polynomials can reveal more about the function's behavior. By factoring, you uncover roots and facilitate further operations like partial fraction decomposition.

A thorough grasp of rational functions allows you to perform integrations involving them more effectively, forming the foundation for more advanced calculus techniques.

Integration Techniques

Integration techniques are methods to find antiderivatives of functions or evaluate definite integrals. They are crucial for solving many problems in calculus. One popular method for evaluating complex integrals is breaking them into simpler parts that are easier to integrate.

• After using long division or factoring, separate the function into simpler terms.



• Standard integration techniques involve power rules, substitution methods, or specific formulas for logarithmic functions.

In this exercise, once the rational function is simplified using partial fractions, each term is integrated separately. Use the known integrals for \(\ln{x}\) and polynomial terms to perform the integration.

Remember that combining techniques like substitution, partial fractions, or integration by parts can further simplify the task, especially for more complicated functions.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. These factors are often linear or quadratic polynomials that multiply back to the original polynomial. The objective is to make expressions easier to handle, especially when they're part of a rational function.

• Factorization helps simplify problems by revealing roots or simplifying the expression into digestible parts.
• In this step, we express \(x^2 - 3x + 2\) as \((x-1)(x-2)\).
• Identifying these factors enables the decomposing of rational functions into partial fractions, simplifying integration.

Once factorization is complete, problems that might initially appear complex transform into exercises where simpler arithmetic or algebraic operations suffice.

Understanding polynomials and their factorization is key to tackling equations and simplifying expressions systematically. It gives you the insight needed to navigate and resolve complex calculus problems.