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Chapter 5: Q. 30 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
${鈭�}_{3}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x$

Short Answer

Expert verified

The value of the integral is ${鈭�}_{3}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x = 1 .$

Step by step solution

01

Step 1. Given information.

The given integral is the following.

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

02

Step 2. Value of integral.

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

solving this improper integral without splitting.

${鈭�}_{3}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x = \lim_{b 鈫� 鈭�} {鈭�}_{3}^{b} {(x - 2)}^{- 2} d x = \lim_{b 鈫� 鈭�} {[- 1 {(x - 2)}^{- 1}]}_{3}^{b} = \lim_{b 鈫� 鈭�} (- {(b - 2)}^{- 1} - (- 1)) = 0 + 1 = 1$

So the value of the integral is ${鈭�}_{3}^{鈭�} \frac{1}{{(x - 2)}^{2}} d x = 1 .$

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Most popular questions from this chapter

Solve the integral: $鈭� \ln (x^{3}) d x$ .

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