   Chapter 13.7, Problem 19E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 1-20, evaluate the improper integrals that converge. 19.   ∫ − ∞ ∞ x 3 e − x 4   d x

To determine

To calculate: The value of the improper integral x3ex4dx if it converges.

Explanation

Given Information:

The provided integral is,

x3ex4dx

Formula used:

According to the exponential rule of integrals,

exdx=ex+C

If the limit defining the improper integral is a unique finite number, then integral converges else it diverges,

f(x)dx=limaacf(x)dx+limbcbf(x)dx

Calculation:

Consider the provided integral,

x3ex4dx

Now, use the formula,

f(x)dx=limaacf(x)dx+limbcbf(x)dx

Multiply and divide by 4 and rewrite the integral as,

x3ex4dx=14[limaa04x3ex4dx+limb0b4x3ex4dx]

Let x4=t, then differentiate both the sides

4x3dx=dt

Thus, the integral becomes,

x3ex4dx=14[limaa0etdt+limb0betdt]

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