# Find p values so that improper integrals converge

Assume that p > 0, find the all the values of p so that the following 2 improper integrals converge :-

$$\int_0^2 \frac{1}{\sqrt{x^{p}-1}} dx$$ $$\int_2^\infty \frac{1}{\sqrt{x^{p}-1}} dx$$

I tried solving them and I got stuck. The first integral must be split into 2 improper integrals between 2 intervals, the first one is between 0 and 1 not and the second one is between 1 and 2 because the integrand is not continuous in 1. I tried substituting variables but nothing worked. Thanks for any help.

HINT

Note that for $x\to 1$

$$x^{p}=(1+(x-1))^p\sim1+p(x-1)\implies \sqrt{x^{p}-1}\sim \sqrt p \sqrt{x-1}$$

then

$$\frac{1}{\sqrt{x^{p}-1}} \sim \frac1{\sqrt p \sqrt{x-1}}$$

and for $x\to \infty$

$$\frac{1}{\sqrt{x^{p}-1}} \sim \frac1{x^{p/3}}$$

• what about x = 1 ? May 25 '18 at 11:25
• Ops yes of course you are right, I update
– user
May 25 '18 at 11:27
• Thanks for the help but could you please explain the situation when x goes to 1 ? it is not very clear to me May 25 '18 at 11:40
• The expression $\frac{1}{(x-1)^{1/3}}$ is integrable. May 25 '18 at 11:53
• I get that the expression that you wrote is integrable but why does the power of x go from p to 1 when x approaches 1 ? May 25 '18 at 11:57