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Chapter 12: Problem 18

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

Short Answer

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

Step by step solution

01

Factorize the denominator

The denominator \(x^{2}-2 x\) can be factorized by taking out a common factor of x. This gives us \(x(x - 2)\).
02

Express the fraction as a sum of partial fractions

Now, express the given function as the sum of partial fractions. This can be done as follows: write \(\frac{2}{x(x - 2)}\) as \(\frac{A}{x} + \frac{B}{x - 2}\). To find the values of A and B, clear the fractions by multiplying both sides by the common denominator \(x(x - 2)\) to get \(2 = A(x - 2) + Bx\). This is an identity, and it should hold for all x. We can choose suitable x values to find A and B. If we let x = 0, the equation becomes 2 = -2A, so A = -1. If we let x = 2, the equation becomes 2 = 2B, so B = 1. Thus, the expression becomes \(-\frac{1}{x} + \frac{1}{x - 2}\).
03

Integrate term-by-term

Now, integrate the terms separately. Remember that the integral of \(\frac{1}{x}\) is \(ln|x|\) and the integral of \(dx\) is x. Then, \(\int \frac{2}{x^{2}-2 x} d x\) becomes \(\int -\frac{1}{x} dx + \int \frac{1}{x - 2} dx\), which evaluates to \(-ln|x| + ln|x - 2| + C\). The C is 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. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$

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. $$ \int_{0}^{1} \frac{1}{x^{2}} d x $$

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.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=4 $$

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 \(n=4 .\).) $$ \int_{0}^{1} e^{-x^{2}} d x $$

Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq \frac{1}{x^{2}}, y \geq 0, x \geq 1 $$
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