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E0: [2GT]Difficulty:*.Fix $n \in ℕ, n \geq 1$ . Let $a_{1}, \dots, a_{n} \in ℝ$ : then

$\begin{aligned} \int_{0}^{a_{1}} & \int_{0}^{a_{2}} \dots \int_{0}^{a_{n}} \cos (x_{1} + x_{2} + \dots x_{n}) d x_{1} d x_{2} \dots d x_{n} = \\ = 2^{n} \cos (\frac{\sum_{i = 1}^{n} a_{i}}{2}) \prod_{i = 1}^{n} \sin (\frac{a_{i}}{2}) \end{aligned}$

$\begin{aligned} \int_{0}^{a_{1}} & \int_{0}^{a_{2}} \dots \int_{0}^{a_{n}} \sin (x_{1} + x_{2} + \dots x_{n}) d x_{1} d x_{2} \dots d x_{n} = \\ = 2^{n} \sin (\frac{\sum_{i = 1}^{n} a_{i}}{2}) \prod_{i = 1}^{n} \sin (\frac{a_{i}}{2}) \end{aligned}$

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Authors: "Mennucci , Andrea C. G." .

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