Chapter 5: Q 29. (page 495)

Convergence and divergence of basic improper integrals: Determine whether each of the given improper integrals converges or diverges. For those that converge, give the exact solution of the integral.
$\int_{1}^{\infty} \frac{1}{x^{p}} d x, p > 1$

Short Answer

Expert verified

The integral is divergent for p>1.

Step by step solution

Step 1. Given information.

Consider the given information.

$\int_{1}^{\infty} \frac{1}{x^{p}} d x$

Step 2. Solve the integral.

Solve the given integral by applying the power rule.

$\begin{array}{rcl} \int_{1}^{\infty} \frac{1}{x^{p}} d x & = & \int_{1}^{\infty} x^{- p} d x \\ = & {[\frac{x^{- p + 1}}{- p + 1}]}_{1}^{\infty} \\ = & \frac{1}{1 - p} [\infty^{- p + 1} - 1^{- p + 1}] \\ = & Undefined \end{array}$

The intergal is divegent for p>1. The integral is divergent for all values of pin the given integral.

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Convergence and divergence of basic improper integrals: Determine whether each of the given improper integrals converges or diverges. For those that converge, give the exact solution of the integral.
$\int_{1}^{\infty} \frac{1}{x^{p}} d x, p > 1$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integral.

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Convergence and divergence of basic improper integrals: Determine whether each of the given improper integrals converges or diverges. For those that converge, give the exact solution of the integral.∫1∞1xpdx,p>1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Pure Maths

Calculus

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Convergence and divergence of basic improper integrals: Determine whether each of the given improper integrals converges or diverges. For those that converge, give the exact solution of the integral.
$\int_{1}^{\infty} \frac{1}{x^{p}} d x, p > 1$