What values of alpha make the integral converge?

[gipc]

Hello,
I need to find the values of alpha for which the following integral converges:
$$\int x^\alpha*ln(x)$$ the integral is from 0 to 1.

I don't really know which test should I use or how to calculate the limit of the integral as x->o+

 

[wwshr87]

Please see attachment for some insight.

 

[gipc]

I still can't see how to find the alphas given that statement after you integrated

 

[wwshr87]

This might not be the correct approach to the problem, I was just trying to give you some insight that perhaps would be helpful. Hopefully someone will clarify this further.

 

[gipc]

I'm sorry for the bump. Can anyone please help?

 

[tmccullough]

It's really all in proceeding from the hint given in the attachment.

You end up with

$$\left(\frac{1}{\alpha+1}x^{\alpha+1}\ln(x)\right)\bigg|_0^1 - \left(\frac{1}{(\alpha+1)^2}x^{\alpha+1}\right)\bigg|_0^1$$

This will exist if (and only if) the two limits

$$\lim_{x\to 0^+}\frac{1}{\alpha+1}x^{\alpha+1}\ln(x),\quad <br /> \lim_{x\to 0^+}\frac{1}{(\alpha+1)^2}x^{\alpha+1}$$

both exist. Use L'Hospital's rule for the first (write it with $$x^{-\alpha-1}$$) on bottom, and the second is immediate. You should get the same answer for both
limits working out.

 

[Susanne217]

Hello,
I need to find the values of alpha for which the following integral converges:
$$\int x^\alpha*ln(x)$$ the integral is from 0 to 1.

I don't really know which test should I use or how to calculate the limit of the integral as x->o+

If you have general integral

$$\int_{a}^{b} f(x) dx$$ and want to find the limit then here is where you do

Assuming that f is continuous on (a,b) and not continuous at x = a.

$$\int_{a}^{b} f(x) dx = \lim_{t \to a^+} \int_{t}^{b} f(x) dx = \lim_{t \to a^+} F(x)|_{t}^{b}$$

 

[gipc]

I understand this steps and everything but I still can't see how to get a specific alphas for which the integral converges. How can I find them?