   Chapter 15.3, Problem 41E

Chapter
Section
Textbook Problem

Use the result of Exercise 40 part (c) to evaluate the following integrals.(a) ∫ 0 ∞ x 2 e − x 2 d x (b) ∫ 0 ∞ x   e − x d x

(a)

To determine

To evaluate: The value of the integral by using the result in exercise 40 part (c).

Explanation

Given:

The function f(x)=x2ex2.

Formula used:

ex2dx=π (1)

Calculation:

Integrate the given integral by using integration by parts taking,

u=xdu=dxdv=xex2dxv=xex2dx.

To integrate v substitute, t=x2,dt=2vdv.

v=xex2dx=12(2)xex2dx=12etdt=12ex2dt

Apply the limit value of x,

v=12[ex2(2x)]

And, by the equation (1),

ex2dx=π20ex2dx=π0ex2dx=π2

Hence, the value of the integral becomes,

0x2ex2dx=[xex22]0(12)0ex2dx=[xex22]0+120ex2dx=[xex22]0+12(π2)=[xex22]0+π4

Apply the limit of x,

0x2ex2dx=lima</

(b)

To determine

To evaluate: The value of the integral by using the result in exercise 40 part (c).

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