/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 27

Evaluate the following integrals or state that they diverge. $$\int_{2}^{\infty} \frac{d x}{(x+2)^{2}}$$

Short Answer

Expert verified
$$\int_{2}^{\infty} \frac{d x}{(x+2)^{2}}$$ Answer: The improper integral converges, and its value is $\frac{1}{4}$.

Step by step solution

01

Identify the given integral as an improper integral

We are given the improper integral: $$\int_{2}^{\infty} \frac{d x}{(x+2)^{2}}$$ Notice that the upper limit of integration is infinity, which signals an improper integral.
02

Rewrite the integral as a limit

Since the integral has infinity as an upper limit, we must rewrite it as a limit to properly evaluate its convergence or divergence: $$\lim_{b \to \infty} \int_{2}^{b} \frac{d x}{(x+2)^{2}}$$
03

Evaluate the antiderivative

We are now going to find the antiderivative of the given integrand, which is: $$\frac{d x}{(x+2)^{2}}$$ Using substitution, let \(u = x + 2\). So, \(\frac{d u}{d x} = 1\) and \(d x = d u\). Now we have: $$\int \frac{d u}{u^2}$$ This is an elementary integral which can be evaluated as follows: $$\int \frac{d u}{u^2} = -\frac{1}{u} + C = -\frac{1}{x + 2} + C$$
04

Apply the Fundamental Theorem of Calculus and take the limit

Now we will apply the Fundamental Theorem of Calculus to evaluate the definite integral and take the limit as the upper limit approaches infinity: $$\lim_{b \to \infty} \left[-\frac{1}{b + 2} - \left(-\frac{1}{2 + 2}\right)\right]$$
05

Evaluate the limit

We will now evaluate the limit: $$\lim_{b \to \infty} \left[-\frac{1}{b + 2} + \frac{1}{4}\right]$$ Since as \(b\) approaches infinity, the first term in the expression goes to zero, we have: $$\lim_{b \to \infty} \left[-\frac{1}{b + 2} + \frac{1}{4}\right] = 0 + \frac{1}{4} = \frac{1}{4}$$ As the limit exists and is finite, the integral converges. Therefore, the value of the integral is: $$\int_{2}^{\infty} \frac{d x}{(x+2)^{2}} = \frac{1}{4}$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Compare the errors in the Midpoint and Trapezoid Rules with \(n=4,8,16,\) and 32 subintervals when they are applied to the following integrals (with their exact values given). \(\int_{0}^{\pi} \ln (5+3 \cos x) d x=\pi \ln \frac{9}{2}\)

Another Simpson's Rule formula is \(S(2 n)=\frac{2 M(n)+T(n)}{3},\) for \(n \geq 1 .\) Use this rule to estimate \(\int_{1}^{e} 1 / x d x\) using \(n=10\) subintervals.

An object in free fall may be modeled by assuming that the only forces at work are the gravitational force and resistance (friction due to the medium in which the object falls). By Newton's second law (mass \(\times\) acceleration \(=\) the sum of the external forces), the velocity of the object satisfies the differential equation $$m \quad \cdot \quad v^{\prime}(t)=m g+f(v)$$, where \(f\) is a function that models the resistance and the positive direction is downward. One common assumption (often used for motion in air) is that \(f(v)=-k v^{2},\) where \(k>0\) is a drag coefficient. a. Show that the equation can be written in the form \(v^{\prime}(t)=\) \(g-a v^{2},\) where \(a=k / m\). b. For what (positive) value of \(v\) is \(v^{\prime}(t)=0 ?\) (This equilibrium solution is called the terminal velocity.) c. Find the solution of this separable equation assuming \(v(0)=0\) and \(0

Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given. $$\int_{0}^{\pi / 2} \ln (\sin x) d x=\int_{0}^{\pi / 2} \ln (\cos x) d x=-\frac{\pi \ln 2}{2}$$

Bob and Bruce bake bagels (shaped like tori). They both make bagels that have an inner radius of 0.5 in and an outer radius of 2.5 in. Bob plans to increase the volume of his bagels by decreasing the inner radius by \(20 \%\) (leaving the outer radius unchanged). Bruce plans to increase the volume of his bagels by increasing the outer radius by \(20 \%\) (leaving the inner radius unchanged). Whose new bagels will have the greater volume? Does this result depend on the size of the original bagels? Explain.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.