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Chapter 12: Problem 25

Use partial fractions to find the indefinite integral. $$ \int \frac{x^{2}-4 x-4}{x^{3}-4 x} d x $$

Short Answer

Expert verified
The indefinite integral of the given function is \(-2\ln|x| - \ln|x-2| + 2\ln|x+2| + C\).

Step by step solution

01

Perform Polynomial Division

First, we need to check if the degree of the numerator \(x^{2}-4 x-4\) is greater than the degree of the denominator \(x^{3}-4 x\). In this case, the degree of the denominator is greater, so, we don't need to perform polynomial division.
02

Decompose into Partial Fraction

Next, we can factorize \(x^{3}-4 x\) into \(x(x^{2}-4)\) or \(x(x-2)(x+2)\). Our fraction becomes: \[\frac{x^{2}-4 x-4}{x(x-2)(x+2)}\]. Then we decompose the fraction into partial fractions. We write it in the form \[\frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2}\] where A, B, and C are constants to be found.
03

Find Constants A, B, and C

Multiply both sides by the denominator \(x(x-2)(x+2)\) to clear the fractions. Now solve for A, B, and C, we get: \(A = -2, B = -1, C = 2\). Therefore, the fraction decomposed into partial fractions is \[= -\frac{2}{x} - \frac{1}{x-2} + \frac{2}{x+2}\].
04

Compute Integral

The next step is to integrate our decomposed fraction. By integrating each term separately, the integral becomes \[-2\int\frac{dx}{x} - \int\frac{dx}{x-2} + 2\int\frac{dx}{x+2}\] which simplifies to \[-2\ln|x| - \ln|x-2| + 2\ln|x+2| + C\]. C is the constant of integration.

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Most popular questions from this chapter

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. $$ \int_{0}^{2}\left(x^{4}+1\right) d x, n=4 $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{2} x^{3} d x $$

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

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=5 $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{-\infty}^{0} e^{2 x} d x $$
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