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Chapter 12: Problem 13

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

Short Answer

Expert verified
The indefinite integral is \(\frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C\).

Step by step solution

01

Express the integrand as the sum of partial fractions

The given integrand is a rational function which can be expressed as the sum of partial fractions. In this case, \(x^2-1\) can be factored into \((x-1)(x+1)\), then the fraction can be decomposed to \(A/(x-1) + B/(x+1)\). By equating this to \(\frac{1}{x^2-1}\), coefficients A and B can be solved.
02

Solve for coefficients A and B

Upon equating, we get \(1 = A(x+1) + B(x-1)\). The coefficients A and B can be solved by letting x=-1 and x=1, and solving for A and B respectively. This gives us: A=1/2 and B=-1/2.
03

Integrate the partial fractions

The integral now becomes two separate integrals: \(\int \frac{1/2}{x-1} dx - \int \frac{1/2}{x+1} dx\), where each fraction can be integrated directly using the formula for the integral of 1/x, which is ln|x|.
04

Write down the final solution

The integration of the two separate fractions gives \(\frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C\), where C stands for the constant of integration.

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Most popular questions from this chapter

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{\infty} e^{-x} d x $$

Marginal Analysis In Exercises 27 and 28, use a program similar to the Simpson's Rule program on page 906 with \(n=4\) to approximate the change in revenue from the marginal revenue function \(d R / d x .\) In each case, assume that the number of units sold \(x\) increases from 14 to 16 . $$ \frac{d R}{d x}=5 \sqrt{8000-x^{3}} $$

Median Age The table shows the median ages of the U.S. resident population for the years 1997 through \(2005 .\) (Source: U.S. Census Bureau) \begin{tabular}{|l|c|c|c|c|c|} \hline Year & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline Median age & \(34.7\) & \(34.9\) & \(35.2\) & \(35.3\) & \(35.6\) \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|c|} \hline Year & 2002 & 2003 & 2004 & 2005 \\ \hline Median age & \(35.7\) & \(35.9\) & \(36.0\) & \(36.2\) \\ \hline \end{tabular} (a) Use Simpson's Rule to estimate the average age over the time period. (b) A model for the data is \(A=31.5+1.21 \sqrt{t}\), \(7 \leq t \leq 15\), where \(A\) is the median age and \(t\) is the year, with \(t=7\) corresponding to 1997 . Use integration to find the average age over the time period. (c) Compare the results of parts (a) and (b).

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{0}^{1}\left(\frac{x^{2}}{2}+1\right) d x, n=4 $$

$$ \int_{1 / 2}^{\infty} \frac{1}{\sqrt{2 x-1}} d x $$
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