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?

In cartography in the year 1599 the following problem arose.  Mercator wanted a map of the world on which compass bearings on the earth correspond to those on the map.  E.g. $13^\circ$ east of north on the surface of the earth corresponds to $13^\circ$ counterclockwise from straight up on the map, at every point.  Going through a bit of geometry, it is found that at latitude $\theta$, the ratio of distance on the map to distance on the earth has to be proportional to $\sec\theta$.  That implies that the location of points on the map at latitude $\theta$ should be at the following distance from the equator:
$$
\int_0^\theta \sec\alpha\,d\alpha.
$$
Evaluating this integral remained a famous unsolved problem for (I think?) more than half a century.
