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Chapter 5: Q. 20 (page 477)

Express each improper integral in Exercises15鈥�20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.
${鈭�}_{2}^{鈭�} \frac{1}{x \ln (x - 2)} d x$

Short Answer

Expert verified

The integral is,

${鈭�}_{2}^{3} \frac{1}{x \ln (x - 2)} d x + \lim_{A 鈫� 鈭�} {鈭�}_{3}^{A} \frac{1}{x \ln (x - 2)} d x .$

Step by step solution

01

Step 1. Given information.

We are given an integrals,

${鈭�}_{2}^{鈭�} \frac{1}{x \ln (x - 2)} d x$

02

Step 2. Graph of function

Let ,

The graph is as follows,

03

Step 3. Expressing the Integral.

The integrand $\frac{1}{x \ln (x - 2)}$ is not defined when $x \ln (x - 2) = 0$

That is $x = 0$ or $\ln (x - 2) = 0$ , which implies $x = 0$ or $3$

So $y = \frac{1}{x \ln (x - 2)}$ is continuous everywhere except at $x = 0, 3$ .

Thus, split the interval $(2, 鈭�)$ at $x = 3$ .

Rewriting the improper integral by splitting the interval $(2, 鈭�)$ at $x = 3$ .

${鈭�}_{2}^{鈭�} \frac{1}{x \ln (x - 2)} d x = {鈭�}_{2}^{3} \frac{1}{x \ln (x - 2)} d x + {鈭�}_{3}^{鈭�} \frac{1}{x \ln (x - 2)} d x$

Write,

${鈭�}_{3}^{鈭�} \frac{1}{x \ln (x - 2)} d x = \lim_{x 鈫� 鈭�} {鈭�}_{3}^{A} \frac{1}{x \ln (x - 2)} d x$

Hence, the integral is,

${鈭�}_{2}^{鈭�} \frac{1}{x \ln (x - 2)} d x = {鈭�}_{2}^{3} \frac{1}{x \ln (x - 2)} d x + \lim_{A 鈫� 鈭�} {鈭�}_{3}^{A} \frac{1}{x \ln (x - 2)} d x .$

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Most popular questions from this chapter

Solve the integral

$鈭� \frac{1}{x^{2} \sqrt{x^{2} - 9}} d x$

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

$(a) 鈭� \frac{4 + x^{2}}{x} d x$ $(b) 鈭� \frac{x}{4 + x^{2}} d x$

role="math" localid="1648759296940" $(c) 鈭� \frac{x^{2}}{4 + x^{2}} d x$ $(d) 鈭� \frac{16 鈭� x^{4}}{4 + x^{2}} d x$

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.

$x^{2} 鈭� 4 x 鈭� 8$

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.

$x^{2} + 6 x 鈭� 2$

Solve $鈭� \frac{x + 2}{{(x^{2} + 4 x)}^{\frac{3}{2}}} d x$ the following two ways:

(a) with the substitution $u = x^{2} + 4 x;$

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

See all solutions

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