   Chapter 5, Problem 3P ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# If ∫ 0 4 e ( x − 2 ) 4 d x = k , find the value ∫ 0 4 x e ( x − 2 ) 4 d x .

To determine

To evaluate: The value of integral function 04xe(x2)4dx.

Explanation

Given information:

The value of the given integral as follows:

04e(x2)4dx=k (1)

Apply integrals of symmetric function as follows:

Show the expression for the function f is even [f(x)=f(x)]

aaf(x)dx=20af(x)dx

Show the expression for the function f is odd [f(x)=f(x)]

aaf(x)dx=0

Calculation:

Consider u=x2 (2)

Differentiate both sides of Equation (2).

u=x2du=dx

Calculate the upper limit for the function u.

Substitute 4 for x in Equation (2).

u(4)=x2=42=2

Calculate the lower limit for the function u.

u(0)=x2=02=2

Apply the upper and lower limits for the function u in Equation (1).

Substitute u for (x2) and du for dx in equation (1).

04e(x2)4dx=22eu4du

Consider f(u)=eu4.

f(u)=eu4f(u)=e(u)4=eu4=f(u) (3)

Compare with the integrals of symmetric function. Hence, f(u) is an even function.

Consider the value of the function g(u)=ueu4.

g(u)=ueu4 (4)

Substitute f(u) for eu4.

g(u)=ueu4=uf(x)

Modify the equation.

g(u)=uf(x)=uf(x)=g(u)

Compare with the integrals of symmetric function

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