   Chapter 7.8, Problem 20E

Chapter
Section
Textbook Problem

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.20. ∫ 2 ∞ y e − 3 y d y

To determine

Whether the integral function is convergent or divergent and evaluate the integral function, if it is convergent.

Explanation

Given:

The integral function is 2ye3ydy.

Definition used:

1. (a) If atf(x)dx exists for every number ta, then

af(x)dx=limtatf(x)dx

Provided this limit exists (as a finite number).

2. (b) If tbf(x)dx exists for every number tb, then

bf(x)dx=limttbf(x)dx

Provided this limit exists (as a finite number).

The improper integrals af(x)dx and bf(x)dx are called convergent if the corresponding limit exists.

The improper integrals af(x)dx and bf(x)dx are called divergent if the corresponding limit does not exist.

3. (c) If both the integral function af(x)dx and af(x)dx are convergent, then

f(x)dx=af(x)dx+af(x)dx

Here, a is any real number.

Calculation:

Apply Part (a) of the definition of improper integral in the given Equation.

2ye3ydy=limt2tye3ydy

Consider the value of the function u=y (1)

Differentiate Equation (1) with respect to y.

dudy=ddy(y)dudy=1du=dy

Consider the value of function dv=e3ydy (2)

Integrate both sides of Equation (2).

dv=e3ydyv=13e3y

Show the formula of integration by parts.

abuv=[uv]ababvdu (3)

Substitute y for u,13e3y for v, dy for du, e3ydy for dv, 2 for a, and t for b in Equation (3).

2ye3ydy=limt2tye3ydy=limt[(y×(13e3y))2t2t(13e3y)dy]=limt[(13ye3y)2t+132te3ydy]=limt[(13ye3y)2t+13(e3y3)2t]

Ȣ

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