Worthwhile first-year-level definite-integrals-"proper" If one mentions the topic of evaluating definite integrals without the fundamental theorem of calculus, I think of things like
$$
\int_0^\infty \frac{\sin x} x \, dx \quad \text{ or } \quad \int_{-\infty}^\infty e^{-x^2}\,dx
$$
i.e. "definite integral[s] proper" defined in this paper as "integral[s] whose value can  be expressed in finite terms, although the indefinite integral of the subject of integration cannot be so determined."
However, at the first-year level, sometimes there is a point to evauating integrals without the fundamental theorem.  For example, one can show by one of the simplest substitutions that
$$
\int_0^{\pi/2} \sin^2\theta\,d\theta = \int_0^{\pi/2} \cos^2\theta\,d\theta, \tag 1
$$
and by a trivial trigonometric identity that their sum is
$$
\int_0^{\pi/2} 1\,d\theta,
$$
and some who post answers here go on to say that that is $\left[\vphantom{\dfrac 1 1}\ \theta\ \right]_0^{\pi/2}$, but that is silly: one is just finding the area of a rectangle.  Thus one evaluates both of $(1)$ without antidifferentiating anything.
Q: What other examples exist of first-year-level integrals doable without antidifferentiating, and where some worthwhile pedagogical purpose can be served by proceeding without antidifferentiating?
 A: I mentioned in the comments how one could similarly calculate the integral of any linear function, but this is rather trivial and does not illustrate why one would want to be able to calculate integrals.
It should also be possible to calculate some less trivial integrals directly via the definition. For example:
$$
\int_0^\alpha x^2dx = \lim_{n \to +\infty} \left( \frac{\alpha}{n} \sum_{k=1}^{n} \left( \frac{\alpha k}{n}\right)^2 \right) = \lim_{n \to +\infty} \left( \frac{\alpha^3}{n^3} \sum_{k=1}^{n} k^2 \right) \\
=\alpha^3\lim_{n \to +\infty}\left(\frac{1}{n^3}\frac{n(n+1)(2n+1)}{6}\right) = \frac{\alpha^3}{3}.
$$
Using linearity of the integral, one can then integrate all polynomials of degree at most $2$. This does use the formula for the sum of the first $n$ integers, which is not trivial (this might motivate the need for a fundamental theorem of calculus).
Similar to the example above, one could exploit the fact that one knows a formula for the partial sums of the geometric series to calculate
$$
\int_0^{\alpha}e^xdx = e^\alpha - 1
$$
but this is calculating the indefinite integral, just without the fundamental theorem of calculus.
