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Chapter 12: Problem 29

Use partial fractions to find the indefinite integral. $$ \int \frac{2 x-3}{(x-1)^{2}} d x $$

Short Answer

Expert verified
The indefinite integral \(\int \frac{2 x-3}{(x-1)^{2}} d x\) is equal to \(2 * ln|x-1| + 1/(x-1) + C \)

Step by step solution

01

Identify the decomposition types

This fraction can be decomposed as \(A/(x-1) + B/(x-1)^2\), where A and B are constants to be found.
02

Equate coefficients on both sides

Multiply each side by \( (x-1)^2 \) to eliminate the denominator. This gives: \(2x -3 = A*(x-1) + B\). This equation must hold for all values of x, so we can choose convenient values of x to find A and B.
03

Find the values of A and B

Set x = 1 to get \(2*1 - 3 = A * (1-1) + B\), which gives B = -1. Substitute x = 0 in \(2x - 3 = A*(x - 1) + B\), we get -3 = -A -1, which gives A = 2.
04

Rewrite the fraction

Now that we have the values of A and B, we can write the fraction as \(2/(x-1) - 1/(x-1)^2\).
05

Carry out the integral

Now we can calculate the integral as \(\int \frac{2}{x-1} dx - \int \frac{1}{(x-1)^2} dx\). This simplifies to \(2 * ln|x-1| + 1/(x-1) + C\), where C is the constant of integration.

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Most popular questions from this chapter

Present Value A business is expected to yield a continuous flow of profit at the rate of \(\$ 500,000\) per year. If money will earn interest at the nominal rate of \(9 \%\) per year compounded continuously, what is the present value of the business (a) for 20 years and (b) forever?

Capitalized Cost In Exercises 51 and 52, find the capitalized cost \(C\) of an asset \((a)\) for \(n=5\) years, \((b)\) for \(n=10\) years, and (c) forever. The capitalized cost is given by \(C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t\) where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance (measured in dollars). [Hint: For part (c), see Exercises \(35-38 .]\) $$ C_{0}=\$ 650,000, c(t)=25,000, r=10 \% $$

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

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. $$ \int_{5}^{\infty} \frac{x}{\sqrt{x^{2}-16}} d x $$
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