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.
Thank you for your help.
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.)
$\displaystyle \frac{3e^{-x} - x^4}{2x^3 + x^5} \to -\frac{1}{x}$
So compute $\displaystyle \int_{10}^{\infty} dx \left ( - \frac{1}{x} \right ) \to -\infty$.
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
$\displaystyle \int_1^{\infty}dx \frac{3e^{-x} - x^4}{2x^3 + x^5} \to \text{finite} - \infty \to -\infty$
Since this answer isn't listed, it must be answer D.
-Dan
For large $\displaystyle x$
$\displaystyle \frac{3e^{-x} - x^4}{2x^3 + x^5} = -O(x^{-1})$
which means that for large $\displaystyle x$ there exist positive constants $\displaystyle k$ and $\displaystyle K$ such that:
$\displaystyle -k\, x^{-1}<\frac{3e^{-x} - x^4}{2x^3 + x^5} < -K\,x^{-1}$
Which is sufficient to show that the integral diverges to $\displaystyle -\infty$, that is the integral does not exist.
RonL
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
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