# “Natural” continuous/smooth non-negative functions $f$ that are unbounded at $\infty$ with finite integral $\int_0^\infty f dx$

It's easy to draw pictures or define piece-wise non-negative functions $f(x)$ that are continuous and unbounded as $x \to \infty$ but have finite indefinite integral $\int_0^\infty f(x) dx$. What are some continuous examples in terms of elementary functions? I was thinking something like $f(x) = x | \cos x|^{x^k}$ might work for some $k \geq 1$ or maybe $f(x) = x | \cos x|^{e^x}$ but I'm having a hard time proving it.

• Sorry, the answer I posted (and deleted) was bounded at $\infty$ - I didn't read your question properly. – Frank Mar 28 '14 at 17:48
• @Frank - I am new here. Does SO allow you to put math symbols or you copy paste them from elsewhere ? Thanks. – Erran Morad Mar 28 '14 at 17:50