Chapter 5: Q. 27 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{1}^{\infty} x^{- 0.99} d x$

Short Answer

Expert verified

The improper integral on $[1, \infty)$ diverges.

$\int_{1}^{\infty} x^{- 0.99} d x = \infty$

Step by step solution

Step 1. Given information.

The given improper integral is following.

$\int_{1}^{\infty} x^{- 0.99} d x$

Step 2. Value of integral.

Write the integral in the proper form of $\int_{1}^{\infty} \frac{1}{x^{p}} d x$

$\int_{1}^{\infty} \frac{1}{x^{p}} d x = \int_{1}^{\infty} x^{- 0.99} d x \int_{1}^{\infty} \frac{1}{x^{p}} d x = \int_{1}^{\infty} \frac{1}{x^{0.99}} d x p = 0.99 & 0.99 < 1$

The Fundamental Theorem of Calculus tells us that the improper integral on $[1, \infty)$ diverges.

$\int_{1}^{\infty} x^{- 0.99} d x = \lim_{b \to \infty} \int_{1}^{b} x^{- 0.99} d x = \lim_{b \to \infty} {[100 x^{0.01}]}_{1}^{b} = \lim_{b \to \infty} (100 b^{0.01} - 100) = \infty$

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Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{1}^{\infty} x^{- 0.99} d x$

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.∫1∞x-0.99dx

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

Decision Maths

Calculus

Logic and Functions

Statistics

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
$\int_{1}^{\infty} x^{- 0.99} d x$