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?
$$
\Phi(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-z^2/2}\,dz
$$

This is the cumulative distribution function of the standard normal distribution, seen in every course on statistics.  Physicists often talk about the "error function", in which $z^2$ appears instead of $z^2/2$ (and then the normalizing constant is different).  That's trivially equivalent to this one, but with this version, the standard deviation is $1$.

Abraham de Moivre considered this function in the 18th century while thinking about a coin-tossing problem.  Suppose you toss a coin 1800 times.  What is the probability that the number of heads is in the range (say) $\{886,\ldots, 908\}$?  An exact answer is computationally far too expensive, but the integral above does it.

The derivative of $\Phi$ is conventionally denoted $\varphi$.

The one seen tabulated in the back of every statistics textbook is $\Phi$.
Why do you need any examples?

In my opinion, FTC is the most important part in Calculus because how it unites the other two parts (dir/int).

The derivative-form of FTC implies that the overall properties of a function (integral) is decided by the domestic properties (derivative).

The integral-form of FTC of FTC indicates that the domestic properties are regulated by the overall properties.

With FTC, it's much easier to do integral with anti-derivatives. It's an example to tell the students how the advancements in mathematics make it easier to solve problems.

I don't think it's needed to make the "first part" of FTC accessible. Actually FTC is FTC and the two so-called parts are essentially the same. They are just represented in two different forms. Similarly, differentiation and integration are essentially the same. So do the two mean value theorems.