Elegance Comes Only Periodically

The question needs re-expressing. The integral is $P(x) \; = \; \int_{-x^2}^0 \prod_{n=1}^\infty \left(1 + \tfrac{t}{n^2}\right)\,dt \; = \; 2 \int_0^x u \prod_{n=1}^\infty \left(1 - \tfrac{u^2}{n^2}\right)\,du$ and hence $P(x) \; = \; 2\int_0^x u \times \frac{\sin \pi u}{\pi u}\,du \; = \; \tfrac{2}{\pi}\int_0^x \sin\pi u\,du \; = \; \tfrac{2}{\pi^2}(1 - \cos \pi x) \;.$ When $x$ is an integer, this integral is either $\frac{4}{\pi^2}$ or zero.

The way the question is written, it wants $P(x)$ to take the nonzero value for all integers $x$ , which is not true. The question should either say "When $x \in \mathbb{Z}$ and $P(x) \neq 0$ ...", or else "When $x \in \mathbb{Z}$ is odd ... " to be accurate, and obtain the intended answer of $4^2 = \boxed{16}$ .