Explanation Definition used: Improper integral Let a function f is continuous on the unbounded interval [ a , ∞ ) . Then the improper integral of f over [ a , ∞ ) is defined as ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x . If it is exist and finite, then it is convergent otherwise divergent. Theorem used: Fundamental Theorem “Let f be a continuous function on [ a , b ] . If F is anti-derivative of f on [ a , b ] , then ∫ a b f ( x ) d x = F ( b ) − F ( a ) ”. Calculation: The given improper integral is ∫ 0 ∞ 1 ( x + 3 ) 2 d x . The integrand f ( x ) = 1 ( x + 3 ) 2 is finite on the unbounded interval [ 0 , ∞ ) . Use the definition and theorem mentioned above to check convergence of the improper integral. ∫ 0 ∞ 1 ( x + 3 ) 2 d x = lim b → ∞ ∫ 0 b 1 ( x + 3 ) 2 d x = lim b

17th Edition

ISBN: 9781323520192

Author: Pearson

Publisher: PEARSON C

See similar textbooks

Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Videos

Question

Chapter 11, Problem 24RE

To determine

To evaluate: The value of improper integral ∫0∞1(x+3)2dx, if it converges.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 23RE

Chapter 11, Problem 25RE

Students have asked these similar questions

What would you say about a set of quantitative bivariate data whose linear correlation is -1? What would a scatter diagram of the data look like? (5 points)

1. Which set of parametric equations is shown in the graph below? Explain your reasoning. a) x = t; y = t² b) x = = t²; y = t -3-2-1 5 4 3 2 1 12 3 2. Using the graph of f, a. determine whether dy/dt is positive or negative given that dx/dt is negative and b. determine whether dx/dt is positive or negative given that dy/dt is positive. Explain your reasoning. 2 f x 1 2 3 4

Find the perimeter of the triangle. Express the perimeter using the same unit of measure that appears on the given sides. 9 ft 13 ft 6 ft The perimeter is

Chapter 11 Solutions

FIN 108 ISU LOOSE >IP<

Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The value of improper integral ∫ 0 ∞ 1 ( x + 3 ) 2 d x , if it converges.

Solution Summary: The author evaluates the value of the improper integral, if it converges, and its value is 13.

Concept explainers

Videos

To evaluate: The value of improper integral ∫0∞1(x+3)2dx, if it converges.

Want to see the full answer?

Chapter 11 Solutions