Question

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it.
integral of the function x/(e^(3x)) from 4 to positive infinity

Answer 1

this is a clac 2 sequence problem by the way

Answer 2

first what is the indefinite integral of x/e^3x ?

Answer 3

-(1/9)(3*x+1)(e^(-3x))

Answer 4

?

Answer 5

Exactly right. So now write down an expression for

\[ \int_4^{R} x e^{-3x} \ dx \]
and then we'll test whether or not the limit of this exists as R --> infinity

Answer 6

ok then what?

Answer 7

Because the integral 4 -> infinity exists by definition if and only if the limit
\[ \lim_{R \rightarrow \infty} \ \ \int_4^R xe^{-3x} \ dx \]
exists.

Answer 8

So to determine whether or not that limit exists, we are going to need an expression for the integral 4 to R

Answer 9

yeah thats where i get confused... i dont know where to go after that step

Answer 10

Well, tell me what the integral 4 to R is.

Answer 11

\[(-(3R+1)/(e^(-3R))9)+(13/((e^-12)9))\]

Answer 12

i dont get why you replace infinity with R

Answer 13

The point is that is the definition of an improper integral, such as this one which goes to infinity

Answer 14

The term in R is
\[- {e^{-3R} \over 9 } (3R + 1)\]
Now the question is does this have a limit as R --> infinity?

Answer 15

The answer is yes and you can find it with l'Hopital's rule by writing this as
\[ { -(3R+1) \over 9e^{3R} } \]

Answer 16

Differentiate top and bottom and the limit will turn out to be zero

Answer 17

THANK YOU SO MUCH!! so the answer would be (13/9)(e^-12) right?

Answer 18

yes.