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Chapter 7: Problem 6

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

Short Answer

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

Step by step solution

01

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.
02

Set Up Partial Fractions

Express \(\frac{x-7}{(x-4)(x+3)}\) as the sum of two fractions: \(\frac{A}{x-4} + \frac{B}{x+3}\).
03

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 \(B = -\frac{10}{7}\).
04

Rewrite the Integral

Substitute back \(A\) and \(B\) into the partial fractions and rewrite the integral: \[\int \left(\frac{-\frac{3}{7}}{x-4} + \frac{-\frac{10}{7}}{x+3}\right) dx\]
05

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:\[\int \frac{x-7}{x^2-x-12} \, dx = -\frac{3}{7}\ln |x-4| - \frac{10}{7}\ln |x+3| + C\]
06

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:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The Law of Mass Action in chemistry results in the differential equation $$ \frac{d x}{d t}=k(a-x)(b-x), \quad k>0, \quad a>0, \quad b>0 $$ where \(x\) is the amount of a substance at time \(t\) resulting from the reaction of two others. Assume that \(x=0\) when \(t=0\). (a) Solve this differential equation in the case \(b>a\). (b) Show that \(x \rightarrow a\) as \(t \rightarrow \infty\) (if \(b>a\) ). (c) Suppose that \(a=2\) and \(b=4\), and that 1 gram of the substance is formed in 20 minutes. How much will be present in 1 hour? (d) Solve the differential equation if \(a=b\).

A tank of capacity 100 gallons is initially full of pure alcohol. The flow rate of the drain pipe is 5 gallons per minute; the flow rate of the filler pipe can be adjusted to \(c\) gallons per minute. An unlimited amount of \(25 \%\) alcohol solution can be brought in through the filler pipe. Our goal is to reduce the amount of alcohol in the tank so that it will contain 100 gallons of \(50 \%\) solution. Let \(T\) be the number of minutes required to accomplish the desired change. (a) Evaluate \(T\) if \(c=5\) and both pipes are opened. (b) Evaluate \(T\) if \(c=5\) and we first drain away a sufficient amount of the pure alcohol and then close the drain and open the filler pipe. (c) For what values of \(c\) (if any) would strategy (b) give a faster time than (a)? (d) Suppose that \(c=4\). Determine the equation for \(T\) if we initially open both pipes and then close the drain.

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2 x^{2}-3 x-36}{(2 x-1)\left(x^{2}+9\right)} d x\)

Evaluate \(\int_{0}^{2 \pi} \frac{x|\sin x|}{1+\cos ^{2} x} d x .\) Hint: Make the substitution \(u=x-\pi\) in the definite integral and then use symmetry properties.

Suppose that the function \(f(x, y)\) depends only on \(x\). The differential equation \(y^{\prime}=f(x, y)\) can then be written as $$y^{\prime}=f(x), \quad y\left(x_{0}\right)=0$$ Explain how to apply Euler's Method to this differential equation if \(y_{0}=0\).
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