Consider the integral
\[ I _ { n } = \int _ { - \infty } ^ { \infty } { \frac { d x } { x } } \sin ( x ) \prod _ { k = 1 } ^ { n } \cos \left( { \frac { x } { 2 k + 1 } } \right) . \]
Show that \(I _ { n } = \pi\) for n = 1, 2, 3, 4, 5, 6. Compute \(I _ { 7 }\)