In my assignment I have to evaluate the (improper) integral, by means of the "comparison theorem". And note whether the function is divergent or convergent. $$\int^{\infty}_{0} \frac{x}{x^3 + 1}dx$$
The comparison theorem basically says
Suppose $f$ and $g$ are continuous functions with $f(x) \geq (x)$ for $x \geq a$. Then:
A) if $\int^\infty_af(x)dx$ is convergent then $\int^\infty_ag(x)dx$ is convergent
B) if $\int^\infty_ag(x)dx$ is divergent then $\int^\infty_af(x)dx$ is divergent
So in other words: to prove if a given integral is convergent you find a function whose integral is larger than the given integral (within the boundaries). And to prove divergence you find a "divergent integral" whose function is always smaller than the function is question.
Now how should I go with this? Is there any trick to the above? Can I "see" (without calculator/automatic plotting) if an integral will be divergent or convergent (so to reduce time)?