Chapter 5: Q. 30 (page 478)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

The given integral is the following.

$\int_{3}^{\infty} \frac{1}{{(x - 2)}^{2}} d x$

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.

$\int_{3}^{\infty} \frac{1}{{(x - 2)}^{2}} d x = \lim_{b \to \infty} \int_{3}^{b} {(x - 2)}^{- 2} d x = \lim_{b \to \infty} {[- 1 {(x - 2)}^{- 1}]}_{3}^{b} = \lim_{b \to \infty} (- {(b - 2)}^{- 1} - (- 1)) = 0 + 1 = 1$

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of integral.

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Use limits of definite integrals to calculate each of the improper integrals in Exercises.∫3∞1x-22dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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