Solve Improper Integration: \int^{\infty}_{0}\frac{x}{(x^{2}+2)^{2}}dx

[Jonathan G]

Homework Statement

$$\int^{\infty}_{0}\frac{x}{(x^{2}+2)^{2}}dx$$

Homework 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.

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.

 

[rock.freak667]

1/(a+b) ≠ 1/a + 1/b


try putting u=x²+2 into the integral.

 

[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.

 

[Jonathan G]

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

Thanks!

 

[Jonathan G]

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

 

[rock.freak667]

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

$$\lim_{x \rightarrow - \infty} e^{ax} =0$$
EDIT: I corrected it, it is $x \rightarrow \infty$

 

[Jonathan G]

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

 

[Mark44]

Homework Statement

$$\int^{\infty}_{0}\frac{x}{(x^{2}+2)^{2}}dx$$

Homework 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.

The Attempt at a Solution

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

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.

Then I separate into 3 different integrals:

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

No, no, no! You really should go back and review how fractions and rational expressions add.

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.