This problem is concerned with evaluating some improper integrals. In particular you will use an improper...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
This problem is concerned with evaluating some improper integrals. In particular you will use an improper integral over an interval of infinite length to evaluate an integral of a function not defined at one end point. This will involve a special function I which arises in many applications in the sciences. a. Evaluate f(log x)² dx. b. Explain how you would evaluate fr8 (log x) dx, but do not actually compute it. Would your method work if the exponent '8' were replaced by '8', or if '7' were replaced by '7'? Why? c. Define the function F(t) = f e-t-¹ dx for t> 0. i. Calculate I(1). ii. Lett be a fixed number, 0 < t < 1. Explain why I (t) is an improper integral. Then, show that the integral that defines I'(t) is convergent. iii. For t > 0, show that I'(t + 1) = tr(t). d. Let n be a positive integer. Use part (c(iii)) to find an equation relating I(n) and n!. Read about mathematical induction and prove your equation is correct. e. Let m be a positive real number and let n be a positive integer. Find an equation relating (log x)" da to I and use this to find a formula for this integral in terms of m and n. f. Use part (e) to evaluate three integrals in (a) and (b). This problem is concerned with evaluating some improper integrals. In particular you will use an improper integral over an interval of infinite length to evaluate an integral of a function not defined at one end point. This will involve a special function I which arises in many applications in the sciences. a. Evaluate f(log x)² dx. b. Explain how you would evaluate fr8 (log x) dx, but do not actually compute it. Would your method work if the exponent '8' were replaced by '8', or if '7' were replaced by '7'? Why? c. Define the function F(t) = f e-t-¹ dx for t> 0. i. Calculate I(1). ii. Lett be a fixed number, 0 < t < 1. Explain why I (t) is an improper integral. Then, show that the integral that defines I'(t) is convergent. iii. For t > 0, show that I'(t + 1) = tr(t). d. Let n be a positive integer. Use part (c(iii)) to find an equation relating I(n) and n!. Read about mathematical induction and prove your equation is correct. e. Let m be a positive real number and let n be a positive integer. Find an equation relating (log x)" da to I and use this to find a formula for this integral in terms of m and n. f. Use part (e) to evaluate three integrals in (a) and (b).
Expert Answer:
Related Book For
Smith and Roberson Business Law
ISBN: 978-0538473637
15th Edition
Authors: Richard A. Mann, Barry S. Roberts
Posted Date:
Students also viewed these computer network questions
-
Managing Scope Changes Case Study Scope changes on a project can occur regardless of how well the project is planned or executed. Scope changes can be the result of something that was omitted during...
-
KYC's stock price can go up by 15 percent every year, or down by 10 percent. Both outcomes are equally likely. The risk free rate is 5 percent, and the current stock price of KYC is 100. (a) Price a...
-
Let F = [F] lb. a) Determine the force Fp acting at roller D. b) Determine the force FE acting at pin E. F 4 ft B 600 lb. ft -3 ft C -3 ft OD -2 ft- E
-
A test rocket is launched by accelerating it along a 200.0-m incline at 1.25 m/s2 starting from rest at point A (Figure 3.45.) The incline rises at 35.0o above the horizontal, and at the instant the...
-
Suppose that f is nonnegative and continuous on [1, 2] and that 21xkf(x)dx = 5 + k2 for k = 0, 1, 2. Prove that each of the following statements is correct. a) (b) c) x2 fx+1)dx 2
-
Petitioner Atlantic Marine Company was a contractor located in Virginia. It entered into a subcontract with J-Crew Management, Inc., a Texas corporation, that contained a forum-selection clause...
-
Retail Inventory Method the records for the Clothing Department of Sharapovas Discount Store are summarized below for the month of January. Inventory, January 1: .............................at...
-
Let G = (N,A) be an undirected graph. (a) Suppose the arcs of G have with nonnegative weights. Explain how you could modify Dijkstra's algorithm (not G) to find a shortest path from a designated node...
-
The following data relate to the operations of Shilow Company, a wholesale distributor of consumer goods: Current assets as of March 31: Cash $ 8,000 Accounts receivable 20,000 Inventory 36,000...
-
The adjusted trial balance for Pharoah Company is given below. PHAROAH COMPANY Trial Balance August 31, 2017 Before After Adjustment Adjustment Dr. Cr. Dr. Cr. Cash $11,860 $11,860 Accounts...
-
Why can the IRR method lead to suboptimal decision making by organizations?
-
When common size ratios are prepared, each asset is compared to ___________ and each expense is compared to _______________. How do common size ratios help compare organizations?
-
How do you evaluate a ratio for a specific year?
-
What is equity financing in the not-for-profit sector?
-
Working capital techniques focus specifically on what aspects of an organizations finances?
-
What are the drawbacks of not agreeing to a lease appearing in the statement of assets.
-
We all experience emotions, but some people disguise their true feelings better than others. Do you think this is a helpful or harmful thing to do? Under what conditions do you think it would be most...
-
Discuss the conditions precedent to the liability of secondary parties.
-
[Northern Corporation entered into a contract with Chugach in August 1966 to repair and upgrade the upstream face of Cooper Lake Dam in Alaska. The contract required Northern to obtain rock from a...
-
Playtime Theaters and Sea-First Properties purchased two theaters in Renton, Washington, with the intention of exhibiting adult films. About the same time, they filed suit seeking injunctive relief...
-
Figure18 . 25 shows the series of Walmart daily closing prices between February 2001 and February 2002 (thanks to Chris Albright for suggesting the use of these data, which are publicly available,...
-
Shipments of Household Appliances. The time plot in Figure 18 . 31 shows the series of quarterly shipments (in million dollars) of US household appliances between 1985 and 1989 (dataare available in...
-
Australian Wine Sales. Figure 18 .32 shows time plots of monthly sales of six types of Australian wines (red, rose, sweet white, dry white, sparkling, and fortified) for 19801994. [Data are available...
Study smarter with the SolutionInn App