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?
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