Convergence test for integral (between -infinty to +infinty)

Question

I have the integral $$\int_{-\infty}^{+\infty}\frac{x+5}{\sqrt{x^6+2x+sinx}}dx$$

I need to detrmine if it converge or diverge, so my idea is to split it for 2 integrals $$\int_{-\infty}^{a}\frac{x+5}{\sqrt{x^6+2x+sinx}}dx\quad +\int_{a}^{+\infty}\frac{x+5}{\sqrt{x^6+2x+sinx}}dx$$

than the second term is easy to show it converge by converge test by using the function $\frac{1}{x^2}$

but I have no idea how to deal with $\int_{-\infty}^{a}\frac{x+5}{\sqrt{x^6+2x+sinx}}dx$
from what I understand that all the converge test is used to apply on $+\infty$

Answer 1

You should also check what happens near $0$. Since, near $0$, the function that you are integrating behaves as $\frac5{\sqrt{3x}}$, you have no problem there.

Otherwise, the method that you've used to chack the convergence at $\infty$ also works for $-\infty$.

Answer 2

The expression under the square root is negative for $x$ between $-1.22...$ and $0$, so the integral is not well-defined. There's a gap where the integrand doesn't exist. I'd call that "divergent" if I had to choose, but a better phrase would be "doesn't exist."