Is there a smooth function with an asymptote at zero and integrable over $]0,\infty[$?

Question

If you look at functions of the form $1/x^k$, $k>0$, it seems you can't have your cake and eat it too. If the integral of $1/x^k$ converges on $[1,\infty[$, then it diverges on $]0,1]$ and vice-versa, and if you try to balance things by picking $1/x$, your greediness is punished by having a function whose integral doesn't converge on both limits.

My question: can you construct a smooth function $f(x)$ that has a vertical asymptote at $x=0$ such that the integral

$$\int_{0}^{\infty} f(x) dx $$

converges?

What if I demand absolute convergence?

Answer 1

you can take $f(x) =e^{-x} .$

Answer 2

Use for example $\frac{1}{\sqrt{x}}e^{-x}$.

Remark: I prefer to note that we can smoothly splice two smooth functions together using a bump function defined over an arbitrarily short interval.