Factoring Integers using Complex Integrals

I can give you a negative answer and a kind of a positive answer to your question. First, similar to what Henry Cohn says, the conventional view of your construction is that it is a restatement of the factoring problem rather than a step to an algorithm. A function with a lot of oscillation is at first glance a function with high information content. So the fact that your setup is numerical rather than discrete doesn't necessarily help anything. In fact, standard numerical integration methods won't look all that different from trial division. Of course it is impossible to rule out some excellent special-purpose numerical integration method, especially because if you apply your transformation backwards, any factoring algorithm is such a special method. And in my opinion subexponential factoring algorithms are a little bit miraculous. You suggest setting up factoring as a 1-dimensional integration problem. But I know of very few numerical analysis algorithms that give a more-than-polynomial gain in speed over obvious algorithms in low dimensions. The main example that comes to mind is spectral methods for solving differential equations or integrals to high accuracy --- but that already starts to look like your construction in reverse.

On the positive side, Shor's quantum algorithm for factoring does look a little bit like your oscillation picture. The algorithm does not look for an oscillation frequency that matches a factor of the number $n$. Instead, it computes the exponents of elements in $(\mathbb{Z}/n)^\times$, which is enough to factor $n$ when $n$ is odd. Using $a^x$ as a periodic function of $x$, it makes a vector $\psi$ in $L^2(\mathbb{Z})$ which is approximately periodic on a large scale with period the exponent of $a$. It then takes an approximate Fourier transform of this vector, and then measures a Fourier mode $k$ weighted according to the square amplitudes of the transform of $\psi$. So this is extracting information from oscillatory integrals! However, crucially, the integral is "computed" with quantum probability, in only the weak sense that a simulated random walk "computes" return probabilities. Quantum amplitudes are NOT stored numbers, they are probability-like state. So their approximate integration in quantum computation is very restricted. However, you have the exotic advantage of handling exponentially many amplitudes, just as randomized algorithms have probabilistic access to exponentially many possibilities.