# Complexity of a specific class of definite integrals

INTRODUCTION: From the answer to this question I learned that deciding whether a definite integral is $$0$$ or not can be NP-complete, as the following integral representation of the Number Partition Problem shows.

Given $$n$$ numbers $$(a_0,a_1,...,a_{n-1})$$ in the range of $$1$$ to $$M$$, find whether the following integral is $$0$$ or not: $$\frac{1}{2\pi}\int_{-\pi}^{\pi} \prod_{k=1}^{n-1} (e^{ia_k \theta} + e^{-ia_k \theta}) d\theta.$$

To see the reduction, notice that the only terms that survive the integration would be the ones where the exponent sum up to $$0$$ and this happens exactly when there is a bipartition of $$(a_0,a_1,...,a_{n-1})$$ where elements of the two parts sum up to the same number. The integral is therefore different from $$0$$ iff such partition exists; this implies that computing a definite integral is at least as hard as Partition, which is known to be NP-complete.

QUESTION: Assuming our machine is able to handle reals, I am now interested in proving (or disproving) the NP-hardness of the following problem

Let $$p(x)$$ be a polynomial on $$\mathbb{R}$$ of degree $$n$$. Decide whether the following integral is $$0$$ or not: $$\int_0^1 \cos(p(x)) dx.$$

More in general, what are the techniques one usually uses in order to do reduction for the problem of finding whether a definite integral is $$0$$ or not on a specific class of functions?

NOTE: This question was also posted on Mathematics Stack Exchange but I guess that attempt was too optimistic.