Functions of the form $\int_a^x f(t) dt$ that are commonly used. I am a graduate student and teaching assistant, and I am teaching Calc 1 for the first time. In a few weeks I will be covering the Fundamental Theorem of Calculus.  I'm using James Stewart's Calculus textbook, and I was hoping to give students several "real world" examples of functions of the form $g(x)=\int_a^x f(t) dt$ to make the first part of the FTC more accessible.  Stewart gives one example, the Fresnel function
$$S(x)=\int_0^x \sin (\pi t^2/2)$$
which is (apparently) used in the theory of the diffraction of light waves.  But I was hoping for more examples from physics, chemistry, etc.  Any thoughts or ideas?

The incomplete Beta function
$$
x\mapsto \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} \int_0^x u^{\alpha-1}(1-u)^{\beta-1}\,du\quad\text{ for }0\le x\le 1
$$
arises in probability and statistics.  When the two parameters $\alpha$, $\beta$ are positive integers, then the incomplete Beta function is a polynomial function, but it probably cannot be expressed more simply than by this integral.
