EDB — 2GT

\begin{align*} ∫_ 0^{a_ 1}& ∫_ 0^{a_ 2}\cdots ∫_ 0^{a_{n}} \cos (x_ 1+x_ 2+\cdots x_ n)\, {\mathbb {d}}x_ 1 \, {\mathbb {d}}x_ 2\cdots \, {\mathbb {d}}x_ n=\\ & =2^{n} \cos \left(\frac{∑_{i=1}^ n a_ i} 2\right) ∏_{i=1}^ n\sin \left(\frac{a_ i}{2}\right) \end{align*}

\begin{align*} ∫_ 0^{a_ 1}& ∫_ 0^{a_ 2}\cdots ∫_ 0^{a_{n}} \sin (x_ 1+x_ 2+\cdots x_ n)\, {\mathbb {d}}x_ 1 \, {\mathbb {d}}x_ 2\cdots \, {\mathbb {d}}x_ n=\\ & =2^ n \sin \left(\frac{∑_{i=1}^ n a_ i} 2\right) ∏_{i=1}^ n\sin \left(\frac{a_ i}{2}\right) \end{align*}