Find c likes the series is convergent

[marrie]

Find c ∈ IR, like ∫∞ ( 2x - c ) dx is convergent
0 X^2 +1 2x+1

I need your help because I was trying to resolve the problem, but I couldn't, is difficult for me.
Please help me!

 

[tiny-tim]

Welcome to PF!

Hi marrie! Welcome to PF!

Find c ∈ IR, like ∫∞ ( 2x - c ) dx is convergent
0 X^2 +1 2x+1

hmm … you mean $$\int_0^{\infty}\left(\frac{2x}{x^2+1} - \frac{c}{2x+1}\right)dx$$

Hint:

i] can you integrate each of them separately? does each converge or diverge?

ii] how much faster do you think the bottom needs to increase than the top for the integral to converge?

 

[marrie]

Ok, if I integrate each of them separately, I believe that the series diverge.
And I know that the bottom needs to increase than for the top for the integral converge.
thanks for your hints!

 

[tiny-tim]

Hi marrie!

Ok, if I integrate each of them separately, I believe that the series diverge.
And I know that the bottom needs to increase than for the top for the integral converge.

ah … but how much more than the top? ​

anyway, write the whole thing over a common denominator …

then what does c have to be to make the top increase slowly enough?