Looking for functions $f$ with $\int_{-\infty}^{\infty}f(x)\,dx = 1$.

Any integrable functions that gives a finite nonzero answer can be modified to suit your need.

Suppose $\int f(x)dx=A$, then let $g(x)=f(x)/A$, automatically we have $\int g(x) dx=A/A=1$.

(Actually, all continuous probability distribution function must have this property.)

This should be a comment but I cannot comment...

The Dirac delta function is not a function!

Any function $f(x)$ which integrates to $1$ over any range $[a,b]$ fits this bill, since we can define $g(x)=f(x)$ on $[a,b]$, and $0$ everywhere else.

Even if you only want continuous functions, restricting ourselves above to $f(x)$ where $f(a)=f(b)=0$ still satisfies this.

If you want continuous functions strictly $>0$ everywhere, these are known as probability distributions (continuous on $[-\infty,\infty]$). A large list of such functions can be found here. A few more notable examples are:

• The normal distribution
• The skew-normal distribution
• The t-distribution
• The cauchy distribution
• The extreme-value distribution