Homework Help: Improper Integration

1. Sep 2, 2009

Jonathan G

1. The problem statement, all variables and given/known data
$$\int^{\infty}_{0}\frac{x}{(x^{2}+2)^{2}}dx$$

2. Relevant equations
I am well aware how it is to be done but when I take a stab at it, I just can't seem to get the correct solution. I think I might be missing a step somewhere or simply starting off incorrect.

3. The attempt at a solution
$$\int ^{\infty}_{0}\frac{x}{(x^{4}+4x+4)}dx$$

Then I separate into 3 different integrals:

$$\int ^{\infty}_{0}\frac{1}{x^{3}}dx$$ + $$\int ^{\infty}_{0}\frac{1}{(4x)}dx$$ + $$\int ^{\infty}_{0}\frac{x}{4}dx$$

and from there I try solving it the rest of the way but I just can't seem to get a solution that I am satisfied with. The first time I got that it diverges, second time i got divided by zero so I'm not sure which 1 to go with if any.

2. Sep 2, 2009

rock.freak667

1/(a+b) ≠ 1/a + 1/b

try putting u=x2+2 into the integral.

3. Sep 2, 2009

Jonathan G

You can't separate the denominator? So I guess it is the numerators you can separate. OK, I'll try this problem again but without expanding the bottom.

4. Sep 2, 2009

Jonathan G

OK i got it. It came out to converging to -1/4

Thanks!

5. Sep 2, 2009

Jonathan G

When you have e^(-2*-infinity) it comes out to e^(infinity) hence infinity?

6. Sep 2, 2009

rock.freak667

$$\lim_{x \rightarrow - \infty} e^{ax} =0$$

EDIT: I corrected it, it is $x \rightarrow \infty$

Last edited: Sep 2, 2009
7. Sep 2, 2009

Jonathan G

What?!? I thought it was if it was as x-> negative infinity =zero : not when x->positive infinity.

Last edited: Sep 2, 2009
8. Sep 2, 2009

Staff: Mentor

You're missing an exponent on one of the terms in the denominator. It should be this:
$$\int ^{\infty}_{0}\frac{x}{(x^{4}+4x^2+4)}dx$$

Actually, you didn't do yourself much good by multiplying it out. You could have directly used the substitution that rock.freak667 suggested.
No, no, no! You really should go back and review how fractions and rational expressions add.