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Chapter 5: Q. 31 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
${鈭�}_{1}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x$

Short Answer

Expert verified

The improper integral ${鈭�}_{1}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x$ diverges.

Step by step solution

01

Step 1. Given information.

The given integral is the following.

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

02

Step 2. Value of integral.

The function $f (x) = \frac{1}{{(x - 2)}^{2}}$ has a vertical asymptote at $x = 2 .$

solve this improper integral by splitting at $x = 2$

role="math" localid="1648878472203" ${鈭�}_{1}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x = {鈭�}_{1}^{2} \frac{1}{{(x - 2)}^{2}} d x + {鈭�}_{2}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x$

Take role="math" localid="1648878379284" ${鈭�}_{1}^{2} \frac{1}{{(x - 2)}^{2}} d x$

role="math" localid="1648878372798" ${鈭�}_{0}^{2} \frac{1}{{(x - 2)}^{2}} d x = \lim_{b 鈫� 0} {鈭�}_{b}^{2} \frac{1}{{(x - 2)}^{2}} d x = \lim_{b 鈫� 0} {[\frac{1}{(x - 2)}]}_{b}^{2} = \lim_{b 鈫� 0} (1 / 0 - \frac{1}{(b - 2)}) = \frac{1}{0} + \frac{1}{2} = undefined$

Integral ${鈭�}_{1}^{2} \frac{1}{{(x - 2)}^{2}} d x$ is undefined and diverges.

So integral ${鈭�}_{1}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x$ is also diverged.

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Most popular questions from this chapter

Solve the integral: $鈭� x \ln \sqrt{x} d x$

Consider the integral $鈭� \sin x \cos x d x$ .

(a) Solve this integral by using u-substitution with $u = \sin x$ and $d u = \cos x d x$ .

(b) Solve the integral another way, using u-substitution with $u = \cos x$ and $d u = 鈭� \sin x d x$ .

(c) How must your two answers be related? Use algebra to prove this relationship.

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.

Explain why it makes sense to try the trigonometric substitution $x = \sec 鈦� u$ if an integrand involves the expression $\sqrt{x^{2} 鈭� 1}$

Find three integrals in Exercises 39鈥�74 that can be solved without using trigonometric substitution.

See all solutions

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