About improper integrals

Question

I have the following function:

$$f(x) = \frac{1}{x^\alpha\sqrt{x^2+1}}$$ and I have to determine the convergence of $\int{}{}_0^{+\infty}f(x)dx$ in relation to $\alpha$

I know that $\int_0^{+\infty}f(x) = \int_0^{1}f(x) + \int_1^{+\infty}f(x)$

Since I didn't understand the problem I looked in the answers and there it read the following:

The first integral converges if $\alpha<1$ since it behaves like $\int_0^{1}\frac{1}{x^\alpha}$.

The second integral converges if $\alpha>0$ since it behaves like $\int_1^{+\infty}\frac{1}{x^{\alpha+1}}$.

Can anyone explain this behaviour please?

Answer 1

Well, whenever $x>1$, you should notice that

$$\sqrt{x^2+1}>x$$

To prove the above, square both sides. It is then noted that

$$\frac1{x^\alpha\sqrt{x^2+1}}<\frac1{x^\alpha\cdot x}=\frac1{x^{\alpha+1}}$$

Thus, by comparison, $\int_1^\infty\frac1{x^\alpha\sqrt{x^2+1}}dx$ converges whenever $\alpha>0$ due to the p-series.

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Whenever $0<x<1$, you should notice that

$$\sqrt{x^2+1}>1$$

Again, just square both sides to see this. It is then noted that

$$\frac1{x^\alpha\sqrt{x^2+1}}<\frac1{x^\alpha}$$

And I assume you can conclude the rest?

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From there, you should test the boundaries. That is, the cases when $\alpha=0,1$.

Answer 2

This is based on the comparison test, one version of which is the following. If there are positive constants $c, C$ such that $0 < c g(x) < f(x) < C g(x)$ for $a < x < b$, then an improper integral $ \int_a^b f(x)\; dx$ converges if and only if $\int_a^b g(x)\; dx$ converges.

If $f(x) = 1/(x^\alpha \sqrt{x^2 + 1})$ and $g(x) = 1/x^\alpha$, we have for $0 < x < 1$, $g(x)/2 < f(x) < g(x)$, so $\int_0^1 f(x)\; dx$ converges if and only if $\int_0^1 (1/x^\alpha)\; dx$ converges, and that is true if and only if $\alpha < 1$.

On the other hand, for $x > 1$ we use

$$ f(x) = \dfrac{1}{x^{\alpha + 1} \sqrt{1 + 1/x^2}}$$ so $$ \dfrac{1}{2 x^{\alpha+1}} < f(x) < \dfrac{1}{x^{\alpha+1}}$$