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Chapter 5: Q. 60 (page 478)

Determine the convergence or divergence of each improper integral in Exercises 57鈥�64 by comparing it to simpler improper integrals whose convergence or divergence is known or can be found directly.
${鈭�}_{1}^{鈭�} \frac{x + 2}{x^{2}} d x$

Short Answer

Expert verified

The integral diverges.

Step by step solution

01

Step 1. Given information.

The given integral is ${鈭�}_{1}^{鈭�} \frac{x + 2}{x^{2}} d x$ .

02

Step 2. Conclusion.

Now,

$鈭� \frac{x + 2}{x^{2}} d x = 鈭� \frac{2}{x^{2}} d x + 鈭� \frac{1}{x} d x = \ln (| x |) - \frac{2}{x} {鈭�}_{1}^{鈭�} \frac{x + 2}{x^{2}} d x = \lim_{8 鈫� 鈭�} {鈭�}_{1}^{n} \frac{x + 2}{x^{2}} d x = \lim_{B 鈫� 鈭�} {(\ln (| x |) - \frac{2}{x})}_{1}^{B} = \lim_{B 鈫� 鈭�} (\ln (| B |) - \frac{2}{B}) - (\ln (| | 鈭�) - \frac{2}{1}) = 鈭� D i v e r g e s i n c o m p a r i s o n w i t h \frac{1}{x}$

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Most popular questions from this chapter

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$2 x^{2} 鈭� 4 x + 1$

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$鈭� \frac{1}{{(x^{2} + 4)}^{\frac{3}{2}}} d x$

Explain why, if $x = a \sin u$ , then $\sqrt{a^{2} 鈭� x^{2}} = a \cos u$ . Your explanation should include a discussion of domains and absolute values.

Solve the integral: $鈭� (3 - x) e^{2 x} d x$

What is a rational function? What does it mean for a rational function to be proper? Improper?

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