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Chapter 12: Problem 20

Use partial fractions to find the indefinite integral. $$ \int \frac{5}{x^{2}+x-6} d x $$

Short Answer

Expert verified
The indefinite integral of the given fraction is \(ln | \frac{x - 2}{x + 3} | + C\)

Step by step solution

01

Factor the Denominator

The denominator of the fraction \(x^{2} + x - 6\) can be factored to \((x-2)(x+3)\). This gives us \(\int \frac{5}{(x-2)(x+3)} dx\)
02

Decompose into Partial Fractions

Decompose the fraction into the form \(\frac{A}{x-2} + \frac{B}{x+3}\). To find the values of A and B, balance the two fractions: \(5 = A(x+3) + B(x-2)\). With x=2, we find A=1. With x=-3, we find B=-1. So, our decomposition is \(\int ( \frac{1}{x-2} - \frac{1}{x+3} ) dx\)
03

Integrate

We use standard formulas to integrate, remembering to put the integral sign in front of each fraction. The integral of \(\frac{1}{x-a}\) is \(ln \lvert x-a \rvert\). Thus our integral is \(ln \lvert x-2 \rvert - ln \lvert x+3 \rvert + C\), where C is our constant of integration. Apply the properties of logarithms to simplify to \(ln \lvert \frac{x-2}{x+3} \rvert + C\)

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Most popular questions from this chapter

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=8 $$

Lumber Use The table shows the amounts of lumber used for residential upkeep and improvements (in billions of board-feet per year) for the years 1997 through \(2005 .\) (Source: U.S. Forest Service) \begin{tabular}{|l|c|c|c|c|c|} \hline Year & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline Amount & \(15.1\) & \(14.7\) & \(15.1\) & \(16.4\) & \(17.0\) \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|c|} \hline Year & 2002 & 2003 & 2004 & 2005 \\ \hline Amount & \(17.8\) & \(18.3\) & \(20.0\) & \(20.6\) \\ \hline \end{tabular} (a) Use Simpson's Rule to estimate the average number of board-feet (in billions) used per year over the time period. (b) A model for the data is $$ L=6.613+0.93 t+2095.7 e^{-t}, \quad 7 \leq t \leq 15 $$ where \(L\) is the amount of lumber used and \(t\) is the year, with \(t=7\) corresponding to 1997 . Use integration to find the average number of board- feet (in billions) used per year over the time period. (c) Compare the results of parts (a) and (b).

Present Value In Exercises 25 and 26, use a program similar to the Simpson's Rule program on page 906 with \(n=8\) to approximate the present value of the income \(c(t)\) over \(t_{1}\) years at the given annual interest rate \(r\). Then use the integration capabilities of a graphing utility to approximate the present value. Compare the results. (Present value is defined in Section 12.1.) $$ c(t)=6000+200 \sqrt{t}, r=7 \%, t_{1}=4 $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{x^{2}} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} x^{2} e^{-x^{3}} d x $$
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