Tani Nevins-Klein

Last year, with grueling effort and a little help from my friend Stewie, I managed to complete my school's Calculus I and II courses. Those classes were perhaps the best math classes I've ever taken, seamlessly weaving intuition with elegant formalism ($\delta$-$\varepsilon$ proofs, anyone?). While my summer math efforts have included tours of number theory, combinatorics, the thing that I have spent perhaps the most time on is evaluating integrals.

The process of evaluating integrals encapsulates everything I love about math: a perfect balance of carefully woven principles and ingenuity. In this post, I would like to present my 3 favorite integrals I've come across these past few months, arranged approximately in ascending order of difficulty.

1. A simple, Fourier-adjacent integral

$$ \int_0^1 \cos(n\pi x) \sin(m \pi x) dx\quad m, n \in \mathbb{Z} $$ I first came across this integral towards the end of the school year, and may be the first “challenge” integral I ever evaluated. After much thought, I managed to rederive the half-remembered product-to-sum identities, and reduce it to a relatively simple form. The part I found tricky at the end was coming up with a closed form expression.

2. A fun series integral

$$ \int_0^1 \left( \sum_{n = 1}^\infty \frac{\lfloor 2^n x \rfloor}{3^n} \right)^2 dx $$ I believe this originally comes from an MIT integration bee problem. I dare not spoil the fun, but it is very helpful to note that $\int_0^1 = \int_0^\frac12 + \int_\frac12^1$.

3. Huh?

$$ \int_1^\infty \left\{ \frac{1}{x} \right\} dx $$

where $\{x\}$ is the fractional part function.

Bonus: a really hard one

$$ \int_{-\infty}^\infty \frac{\log \left| \zeta (1/2 + it) \right|}{1 + 4t^2} dt $$

Email me if you solve this one.