Test integral for convergence

Question

I have to test for which values of $y$ the following integral converges:

$$ \int_{0}^{\infty}\frac{\arctan\left(x^{3}\right)\ln\left(1 + x^{2}\right)}{x^{y}} \,\mathrm{d}x $$

I have tried some different ways, but I failed. Any hints ?.

Answer 1

The integrand is a continuous function over $(0,\infty)$, potential issues are as $x \to 0^+$ and as $x \to \infty$.

• As $x \to 0^+$, on gets $$ \frac{\arctan\left(x^{3}\right)\ln\left(1 - x^{2}\right)}{x^{y}} \sim -x^{5-y} $$ giving a convergent integral iff $y-5<1$.
• As $x \to \infty$, on gets $$ \frac{\arctan\left(x^{3}\right)\ln\left(1 + x^{2}\right)}{x^{y}} \sim \pi \,\frac{\ln x}{x^y} $$ giving a convergent integral iff $y>1$.

Then you can conclude.

Answer 2

**hint near 0 **

The integrand is equivalent to

$$\frac {x^3.x^2}{x^y}\sim \frac{1}{x^{y-5}} $$

so, near 0, it converges if $$y-5<1\iff <6.$$