I'm interested in the potential of such a technique. I got the idea from Moron's answer to [this question](https://math.stackexchange.com/questions/9402/9409#9409), which uses the technique of differentiation under the integral.

Now, I'd like to consider this integral:

$$\int\_{-\pi}^\pi \cos{(y(1-e^{i\cdot n \cdot t}))}\mathrm dt$$

I'd like to differentiate with respect to y. This will give the integral:

$$\int\_{-\pi}^\pi -(1-e^{i\cdot n \cdot t})(\sin{(y(1-e^{i\cdot n \cdot t}))}\mathrm dt$$

...If I'm correct. Anyways, I'm interested in obtaining the results to this second integral, using this technique. So I'm wondering if solving the first integral can help give results for the second integral. I'm thinking of setting $y=1$ in the second integral. This should eliminate $y$ from the result, and give me the integral involving $x$.

The trouble is, I'm not sure I can use the technique of differentiation under the integral. I want to know how I can apply this technique to the integrals above. Any pointers are appreciated.

For instance, for what values of $y$ is this valid?