Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x-7}{x^{2}-x-12} d x\)

Short answer

The integral is \(-\frac{3}{7}\ln |x-4| - \frac{10}{7}\ln |x+3| + C\).

Step-by-step solution

Factor the Denominator

First, we need to factor the quadratic denominator. The expression \(x^2 - x - 12\) can be factored as \((x - 4)(x + 3)\). This will allow us to decompose the fraction into partial fractions.

Set Up Partial Fractions

Express \(\frac{x-7}{(x-4)(x+3)}\) as the sum of two fractions: \(\frac{A}{x-4} + \frac{B}{x+3}\).

Solve for Coefficients A and B

Multiply both sides by the denominator \((x-4)(x+3)\) to clear the fractions: \[x - 7 = A(x+3) + B(x-4)\]Set up equations by substituting convenient values: - For \(x = 4\), solve \(4 - 7 = A(4 + 3)\) to find \(A = -\frac{3}{7}\).- For \(x = -3\), solve \(-3 - 7 = B(-3 - 4)\) to find \(B = -\frac{10}{7}\).

Rewrite the Integral

Substitute back \(A\) and \(B\) into the partial fractions and rewrite the integral: \[\int \left(\frac{-\frac{3}{7}}{x-4} + \frac{-\frac{10}{7}}{x+3}\right) dx\]

Integrate Each Term

Integrate each fraction separately: - The integral of \(\frac{-\frac{3}{7}}{x-4}\) is \(-\frac{3}{7} \ln |x-4|\).- The integral of \(\frac{-\frac{10}{7}}{x+3}\) is \(-\frac{10}{7} \ln |x+3|\).Therefore, the integral becomes:\[\int \frac{x-7}{x^2-x-12} \, dx = -\frac{3}{7}\ln |x-4| - \frac{10}{7}\ln |x+3| + C\]

Combine the Logarithms

Combine the logarithm terms to simplify the expression: \[-\frac{3}{7}\ln |x-4| - \frac{10}{7}\ln |x+3| = \ln \left(\frac{(x-4)^{-3/7}}{(x+3)^{10/7}}\right) = \ln \left(\frac{|x-4|^{3/7}}{|x+3|^{10/7}}\right)\]This gives us the final answer for the integral:

Key concepts

Integration Techniques

Integration techniques are the methods used to find the integral of a function. When dealing with complex expressions, these techniques become essential. In this specific exercise, we use the method of partial fraction decomposition. This technique breaks a complex fraction into simpler parts, making integration easier.

Partial fraction decomposition is particularly useful for integrating rational functions. Once a function is split into simpler fractions, basic integration becomes possible. The simpler fractions often lead to logarithmic functions post-integration.

Therefore, mastering different integration techniques, like partial fraction decomposition, broadens your tools, allowing you to handle a variety of integral calculus problems smoothly.

Rational Functions

Rational functions are expressions formed as the quotient of two polynomials. In this exercise, we're working with the rational function \(\frac{x-7}{x^2-x-12}\).

When dealing with rational functions, it's important to consider both the numerator and the denominator. Often, the denominator can be factored, which is key to applying partial fraction decomposition. In our example, the denominator \(x^2-x-12\) is factorable into \((x-4)(x+3)\).

The partial fraction decomposition then expresses the entire rational function as simpler fractions added together, making it straightforward to integrate each individually. Understanding how to break down these functions is crucial in integral calculus.

Integral Calculus

Integral calculus is a branch of mathematics focused on finding integrals. Integrals are used to calculate areas, volumes, central points, and many other concepts. The fundamental theorem of calculus links the process of differentiation with integration, essentially reversing the differentiation operation.

In our exercise, we're finding the indefinite integral of a rational function. Indefinite integrals result in a family of functions that differ by a constant. This constant, usually denoted as \(C\), represents the antiderivative's family.

The use of logarithmic functions in integration is common, especially when dealing with partial fractions. This exercise demonstrates how such integrals lead naturally to logarithmic expressions.

Factorization

Factorization is the process of breaking down an expression into a product of simpler factors. It's a critical step when dealing with rational functions and partial fractions, as seen in this exercise.

Here, the factorization of the quadratic expression \(x^2-x-12\) into \((x-4)(x+3)\) is crucial. This step enables the decomposition of the original rational function into simpler fractions.

Knowing how to factor efficiently and accurately can often simplify complex calculus problems. It turns a challenging problem into a manageable one by making decomposition and subsequent integration much more straightforward.

• Practice recognizing factorable expressions.
• Use factoring to identify points of decomposition.
• Factorization is often the first key step in solving and simplifying integrals.