Explanation Given information: ∫ − ∞ ∞ e − x ( e − x + 2 ) 2 d x Explanation: Definition 1 : Let a be fixed and suppose that f ( x ) is a nonnegative function for x ≥ a and if lim b → ∞ ∫ a b f ( x ) d x = L , then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x = L . Definition 2 : Let b be fixed and suppose that f ( x ) is a nonnegative function for x ≤ b and if lim a → − ∞ ∫ a b f ( x ) d x = L , then ∫ − ∞ b f ( x ) d x = lim a → − ∞ ∫ a b f ( x ) d x = L . The improper integral ∫ − ∞ ∞ f ( x ) d x can be written as ∫ − ∞ 0 f ( x ) d x + ∫ 0 ∞ f ( x ) d x . Therefore, ∫ − ∞ ∞ e − x ( e − x + 2 ) 2 d x = ∫ − ∞ 0 e − x ( e − x + 2 ) 2 d x + ∫ 0 ∞ e − x ( e − x + 2 ) 2 d x By using definition 1 and 2 , ∫ − ∞ 0 e − x ( e − x + 2 ) 2 d x + ∫ 0 ∞ e − x ( e − x + 2 ) 2 d x = lim a → − ∞ [ ∫ a 0 e − x ( e − x + 2 ) 2 d x ] + lim b → ∞ [ ∫ 0 b e − x ( e − x + 2 ) 2 d x ] Now solve improper integrals by using substitution method as follows: u = e − x + 2 ⇒ d u = − e − x d x ⇒ − d u = e − x d x When x = 0 , u = 3 ; when x = a , u = e − a + 2 . Substitute all these values in ∫ a 0 e − x ( e − x + 2 ) 2 d x , it gives ∫ a 0 e − x ( e − x + 2 ) 2 d x = ∫ e − a + 2 3 − 1 u 2 d u By integrating, = 1 u | e − a + 2 3 = 1 3 − 1 e − a + 2 As a → − ∞ , 1 e − a + 2 approaches to 0

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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 9.6, Problem 44E

To determine

The value of the improper integral ∫−∞∞e−x(e−x+2)2 dx, whenever it is convergent.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.6, Problem 43E

Chapter 9.6, Problem 45E

Students have asked these similar questions

An engineer is designing a pipeline which is supposed to connect two points P and S. The engineer decides to do it in three sections. The first section runs from point P to point Q, and costs $48 per mile to lay, the second section runs from point Q to point R and costs $54 per mile, the third runs from point R to point S and costs $44 per mile. Looking at the diagram below, you see that if you know the lengths marked x and y, then you know the positions of Q and R. Find the values of x and y which minimize the cost of the pipeline. Please show your answers to 4 decimal places. 2 Miles x = 1 Mile R 10 miles miles y = miles

An open-top rectangular box is being constructed to hold a volume of 150 in³. The base of the box is made from a material costing 7 cents/in². The front of the box must be decorated, and will cost 11 cents/in². The remainder of the sides will cost 3 cents/in². Find the dimensions that will minimize the cost of constructing this box. Please show your answers to at least 4 decimal places. Front width: Depth: in. in. Height: in.

Find and classify the critical points of z = (x² – 8x) (y² – 6y). Local maximums: Local minimums: Saddle points: - For each classification, enter a list of ordered pairs (x, y) where the max/min/saddle occurs. Enter DNE if there are no points for a classification.

Chapter 9 Solutions

Calculus & Its Applications (14th Edition)

Knowledge Booster

$Calculus$

Learn more about

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

The value of the improper integral ∫ − ∞ ∞ e − x ( e − x + 2 ) 2 d x , whenever it is convergent.

Solution Summary: The author analyzes the value of the improper integral displaystyle 'underset-infty', whenever it is convergent.

Concept explainers

Videos

The value of the improper integral ∫−∞∞e−x(e−x+2)2 dx, whenever it is convergent.

Want to see the full answer?

Chapter 9 Solutions