Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. $$ \int_{-1}^{2} \frac{x}{x^{2}-9} d x $$

Step-by-Step Solution

Answer

The solution to the integral is \( ln(2.5) \). Depending on the accuracy of the symbolic integration utility, the results should be similar with possible differences only due to rounding or approximation errors.

Step 1: Fraction Decomposition

Perform fraction decomposition on the integrand \( \frac{x}{{x^{2} - 9}} \), which can be rewritten as \( \frac{x}{{(x - 3)(x + 3)}} \). The fraction can be decomposed into \( A/(x - 3) + B/(x + 3) \). Multiplying through by the denominator gives us x = A(x + 3) + B(x - 3). Setting x = 3 gives A = 3/6 = 1/2, and setting x = -3 gives B = -3/6 = -1/2.

Step 2: Integral Calculation

The integral from -1 to 2 of the fraction is then divided into two separate integrals, \( \int_{-1}^{2} \frac{1/2}{x-3} dx - \int_{-1}^{2} \frac{1/2}{x+3} dx \), then find the antiderivative of each. For \( \int_{-1}^{2} \frac{1/2}{x-3} dx \), it's \( \left[1/2 * ln|x-3|\right]_{-1}^{2} \) = \( ln|2-3| - ln|-1-3| \). For \( \int_{-1}^{2} \frac{1/2}{x+3} dx \), it's \( \left[-1/2 * ln|x+3|\right]_{-1}^{2} \) = \( -ln|2+3| + ln|-1+3| \).

Step 3: Calculate the Differences

Now calculate the differences of the two calculated results. It equals \( (ln|1| - ln|4|) - (-ln|5| + ln|2|) \) = -ln(4) + ln(5) - ln(2) = -ln(4/5) + ln(2) = ln(\( \frac{2}{4/5} \)) = ln(2.5).

Key Concepts

Fraction Decomposition

To solve the problem of integrating \( \frac{x}{x^2-9} \), we start with fraction decomposition. The integrand \( \frac{x}{x^2-9} \) can be factored into \( \frac{x}{(x-3)(x+3)} \). This transformation allows us to express it as a sum of two simple fractions.
We write it as \( \frac{A}{x-3} + \frac{B}{x+3} \). By clearing the denominators, we equate \( x = A(x+3) + B(x-3) \).
To find the values of \( A \) and \( B \), we set convenient values for \( x \). When \( x = 3 \), it simplifies to \( A = 1/2 \). Similarly, when \( x = -3 \), \( B = -1/2 \).
This strategic decomposition makes integration manageable by turning a complicated fraction into simpler parts.

Antiderivative Calculation

Once we decompose the fraction, the next step is to calculate the antiderivative. We compute it separately for each term.
The integral is split into two parts: \( \int \frac{1/2}{x-3} \, dx \) and \( -\int \frac{1/2}{x+3} \, dx \).

• For the first integral, the antiderivative is \( \frac{1}{2} \ln |x-3| \).
• For the second integral, we have \( -\frac{1}{2} \ln |x+3| \).

Each term uses the natural logarithm due to the \( \frac{1}{x} \) form.
In this way, calculating the antiderivatives becomes simpler and straightforward.

Symbolic Integration

Symbolic integration involves using various tools or software to perform integration. These systems utilize mathematical algorithms to find antiderivatives symbolically rather than numerically.
This approach can sometimes show results in different forms due to simplifications or alternate methods used within the software. Differences might arise in constant terms or sign conventions.
For the integral \( \int \frac{x}{x^2-9} \, dx \), symbolic integrators handle the decomposition and logarithmic returns automatically. They might offer direct solutions that need interpretation but result in equivalent expressions in terms of logarithmic functions.

Logarithmic Integration

Logarithmic integration appears when integrands take the form of \( \frac{1}{x} \). This leads to a natural logarithm in the antiderivative.
For our original fractions, we apply this principle:

• \( \int \frac{1/2}{x-3} \, dx = \frac{1}{2} \ln |x-3| \).
• \( \int \frac{-1/2}{x+3} \, dx = -\frac{1}{2} \ln |x+3| \).

The definite integral limits result in the difference of log values, simplifying the evaluation process.
Understanding logarithmic integration is essential for recognizing and handling integrals involving similar rational fractions.