Worthwhile freshman-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 freshman 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 freshman-level integrals doable without antidifferentiating, and where some worthwhile pedagogical purpose can be served by proceeding without antidifferentiating?