Math Help - integral

1. integral

the improper integral from 1 to + infinity

3e^(-x) - x^(4)
______________

2x^(3) + x^(5)

A)diverges to + inf
B)exists and is finite
C)doesn't exist
D)none of the preceding

The right answer is D but I can't see why.

2. Originally Posted by 0123
the improper integral from 1 to + infinity

3e^(-x) - x^(4)
______________

2x^(3) + x^(5)

A)diverges to + inf
B)exists and is finite
C)doesn't exist
D)none of the preceding

The right answer is D but I can't see why.
I'm not quite sure what the definition of "doesn't exist" is supposed to be because any integral that is (+/-) infinity doesn't exist, so A and C should be the same!

Regardless, look at the form of the integrand for large x (which translates in this case for any x greater than 5 - 10 or so.)
$\frac{3e^{-x} - x^4}{2x^3 + x^5} \to -\frac{1}{x}$

So compute $\int_{10}^{\infty} dx \left ( - \frac{1}{x} \right ) \to -\infty$.

This by itself is not an answer, but if you sketch the integrand over [1, 10] you will see that the area under the curve is clearly finite. Thus
$\int_1^{\infty}dx \frac{3e^{-x} - x^4}{2x^3 + x^5} \to \text{finite} - \infty \to -\infty$

-Dan

3. Originally Posted by topsquark
I'm not quite sure what the definition of "doesn't exist" is supposed to be because any integral that is (+/-) infinity doesn't exist, so A and C should be the same!

Regardless, look at the form of the integrand for large x (which translates in this case for any x greater than 5 - 10 or so.)
$\frac{3e^{-x} - x^4}{2x^3 + x^5} \to -\frac{1}{x}$

So compute $\int_{10}^{\infty} dx \left ( - \frac{1}{x} \right ) \to -\infty$.

This by itself is not an answer, but if you sketch the integrand over [1, 10] you will see that the area under the curve is clearly finite. Thus
$\int_1^{\infty}dx \frac{3e^{-x} - x^4}{2x^3 + x^5} \to \text{finite} - \infty \to -\infty$

-Dan
For large $x$

$\frac{3e^{-x} - x^4}{2x^3 + x^5} = -O(x^{-1})$

which means that for large $x$ there exist positive constants $k$ and $K$ such that:

$-k\, x^{-1}<\frac{3e^{-x} - x^4}{2x^3 + x^5} < -K\,x^{-1}$

Which is sufficient to show that the integral diverges to $-\infty$, that is the integral does not exist.

RonL

4. Firstly, thank you both. Then, I am getting a little bit confused..ok, now, I sketched the graph and clearly it shows the area is finite, but without that there are still some points that I don't get (sorry for annoying you.. ) I arrived 'till (-1/x), since qualitatively we can say the integrand will be like that, but then I am lost. As far as I remember, 1/x diverges and if something diverges then the area is infinite...which is surely not correct, otherwise the answer would have been another...help!
thanks thanks thanks so much

5. Originally Posted by 0123
Firstly, thank you both. Then, I am getting a little bit confused..ok, now, I sketched the graph and clearly it shows the area is finite
How does the sketch show the area to be finite?

Just because the absolute value of the integrand is decreasing as x increases does not make the area finite.

RonL

6. Originally Posted by CaptainBlack
How does the sketch show the area to be finite?

Just because the absolute value of the integrand is decreasing as x increases does not make the area finite.

RonL
The integral is finite over the interval [1, 10]. That is what I was referring to.

-Dan

Edit: The way I replied to this makes it look like I was replying to CB. My response was meant for 0123. Sorry about that!

7. sorry.. I got it, thanks!!!

8. Originally Posted by 0123
well the book says "none of the preceding", not "doesn't exist"... and well, what did I(=the program) sketched since I came out with a different graph
so, now...I don't understand..
The point is that both your integrand and -1/x share similar characteristics at large x. This is enough to say that both integrals are infinite. CaptainBlack's graph showed what the integrand looks like for x > 0. My own graph confirms his. I'm not sure what your graph looks like. (Unless you sketched it for all x, in which case it does look rather different. But remember all we need is the interval [1, infinity). )

-Dan

9. Originally Posted by topsquark
The integral is finite over the interval [1, 10]. That is what I was referring to.

-Dan

Edit: The way I replied to this makes it look like I was replying to CB. My response was meant for 0123. Sorry about that!
Sorry about the confusion, I hadn't realised that 0123 was referring to
what you wrote (which of course was right)

RonL

10. sorry also on my part, I created much of this confusion. and thanks again

11. Originally Posted by CaptainBlack
Sorry about the confusion, I hadn't realised that 0123 was referring to
what you wrote (which of course was right)

RonL
(snort) Lately what I've been writing distinctly does NOT fall under the category "which of course was right." I'd say I need a vacation, but I just got off one!

-Dan