I have the following integral:
$$I=\int_0^\infty\frac{x}{x^5+5}dx$$
I have to use the comparison theorem to tell if it's convergent or divergent.
I have tried using $\frac{x}{x^5}=\frac{1}{x^4}$ as a function to compare I with, but $\int_0^\infty\frac{1}{x^4}dx$ is divergent and bigger than I: therefore I cannot make any conclusion about I. I could use $\frac{1}{x^5+5}$ as a function to compare, but I don't know how to integrate that, and it is only similar to I at $\infty$, not at $0$.
How should I tell if it's convergent or divergent?
Best Answer
Comparison with $\frac{1}{x^4}$ is the right way of doing this. Your integral is only improper at its upper boundary, and so the convergence there does not depend on the lower boundary: you could just as well test the convergence of the integral:
$$\int_c^{\infty}\frac{x}{x^5+5}dx$$
for some $c>0$ (e.g. $c=1$) - for which the function $\frac{1}{x^4}$ can be used. Due to convergence of:
$$\int_c^{\infty}\frac{dx}{x^4}$$
the original integral also converges.