Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Writing State the method you would use to evaluate each integral. Explain why you chose that method. (Do not integrate.) (a) \(\int \frac{2 x+1}{x^{2}+x-8} d x\) (b) \(\int \frac{7 x+4}{x^{2}+2 x-8} d x\)

Step-by-Step Solution

Answer

For the first integral, the method of partial fractions could be used as the numerator can be written as the linear combination of the derivatives of the denominator. For the second integral, one would still use partial fraction decomposition, but long division would be necessary first.

Step 1: Analyzing Integral (a)

Looking at the first integral, \(\int \frac{2 x+1}{x^{2}+x-8} d x\), an individual can see that the denominator can be factored where \(x^{2}+x-8 = (x+4)(x-2)\). Also, the numerator can be written as a linear combination of the derivative of the two factors of the denominator as follow: \(2x+1 = 1*(-4-(-2)) + 2*(4-2)\). So the method of choice here would be the method of partial fractions.

Step 2: Analyzing Integral (b)

Looking at the second integral, \(\int \frac{7 x+4}{x^{2}+2 x-8} d x\), one can see that the denominator can also be factored where \(x^{2}+2x-8 = (x+4)(x-2)\). However, the numerator cannot be written as a combination of the derivatives of these terms. So, the most likely method to solve this integral would be long division followed by the method of partial fractions.

Key Concepts

Method of Partial Fractions

The Method of Partial Fractions is an essential algebraic tool in calculus that allows us to integrate rational functions, which are defined as the quotient of two polynomials. When faced with a complex fraction that's difficult to integrate directly, partial fractions can simplify the fraction into a sum of simpler fractions.

For an integral like \(a\), \( \int \frac{2 x+1}{x^{2}+x-8} d x \), we notice that the denominator can be factored into \( (x+4)(x-2) \)—this is a perfect setup for partial fractions. This method works by expressing the complex fraction as a sum of simpler terms like \( \frac{A}{x+4} + \frac{B}{x-2} \), where \(A\) and \(B\) are constants that we need to solve for. Once we find these constants, each term in the sum can be integrated more easily. The algebra involved often includes setting up a common denominator and equating coefficients of like terms to solve for these unknowns.

For students tackling this type of problem, it is vital to get comfortable with factoring and with the process of finding these constants, as they are key steps in the method of partial fractions.

Algebraic Long Division

Algebraic long division is a process similar to long division with numbers but is used for dividing polynomials. It is especially useful when the degree of the numerator is equal to or greater than the degree of the denominator in a rational expression.

For example, integral \(b\), \( \int \frac{7 x+4}{x^{2}+2 x-8} d x \), leads us to perform algebraic long division because we cannot directly apply partial fractions. We can think of long division as a way to simplify the expression so that we end up with a polynomial plus a remainder over the original divisor. This remainder can then often be further decomposed using partial fractions if necessary.

Long Division Steps

• Divide the first term of the numerator by the first term of the denominator and write the result above the long division bar.
• Multiply the divisor by this result and subtract it from the numerator.
• Bring down the next term of the numerator and repeat the process until you have a remainder with a degree less than that of the divisor.

After division, students are typically left with an easier integral to solve, either a polynomial or a simpler rational function that may require additional techniques.

Integral Evaluation Methods

Integral evaluation methods are a collection of techniques used to calculate the definite or indefinite integral of a function. A few common methods include integration by substitution, integration by parts, and trigonometric integration in addition to the aforementioned partial fractions and algebraic long division.

When deciding which method to use for evaluating an integral, one must examine the integrand's structure. If the function is a simple polynomial, basic integration rules apply. For products of functions, integration by parts is appropriate. Integration by substitution can be used when a function's derivative is present in the integrand. For integrals involving trigonometric functions, identities or trigonometric substitution could be the way to go.

It is essential for students to understand that sometimes more than one method can apply, and part of the skill in calculus is determining the most efficient strategy. For example, a combination of strategies might be necessary, such as first performing long division and then using partial fractions, as seen in integral \(b\). Developing this intuition often requires practice and familiarity with a variety of functions and their properties.