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?

• I presume that the formula for integrals of the form $\int_{x_1}^{x_2} (ax + b)$ can be derived similarly from the formula of the area of a trapezoid, but you're looking for less 'trivial' integrals, I presume? – Bib-lost May 26 '16 at 20:43
• How about the Borwein integrals? – anomaly May 26 '16 at 20:44
• Today there was $\int_0^3 \sqrt{9-x^2}\,dx$. One could also look at relatives, such as the integral from $0$ to $3/2$. – André Nicolas May 26 '16 at 20:48
• Are you looking for examples along the lines of recognizing $\int_{-a}^a \sqrt{a^2 - x^2}\ dx$ as a semicircular area, or that $\int_{\pi/2}^{3\pi/2}\sin x\ dx$ is odd about the center of the interval? Or something a little more subtle? Freshman aren't too well acquainted with exploiting orthogonality in integration, but that's another idea... I think I'm unclear about how the sinc and Gaussian examples connect to the question in your post. – zahbaz May 26 '16 at 20:52

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.
• ok, I confess that the question as I state it does not exclude this sort of thing, but it has a different flavor from what I had in mind. Lots of calculus books mention limits of Riemann sums, and some even say that Archimedes used those (his actual methods (he had at least two of them) are far better). $\qquad$ – Michael Hardy May 26 '16 at 23:08