Comparing two almost identical improper integrals I saw a similar problem in my calculus textbook, which got me thinking. I can see why $\int_{-\infty}^01/x^3$ doesn't converge, but how come a very similar function, $\int_{-\infty}^01/\sqrt (x^6+1)$ does converge? 

Because, unlike $\dfrac{1}{x^3}$, the function $\dfrac{1}{\sqrt{x^6+1}}$ behaves nicely near $0$. Indeed, the second function is continuous in the interval $[-1,0]$. 
