$I_n(x)$ is defined as the following.
$$ I_n(x) := \int_0^{\infty } \left(\prod _{k=1}^n \frac{\sin \left(\displaystyle\frac{t}{ k^x}\right)}{\displaystyle\frac{t}{k^x}}\right) \, \mathbb{d}t$$
We know
$$ I_1(1) = I_2(1) = I_3(1) = \frac{\pi}{2},$$ $$ I_4(1) = \frac{1727 \pi}{3456}, I_5(1) = \frac{20652479 \pi}{41472000},$$ $$ I_6(1) = \frac{2059268143 \pi}{4147200000}, I_7(1) = \frac{24860948333867803 \pi}{50185433088000000}, \cdots .$$
Now, prove
$$ I_n(x) = \frac{\pi}{2}$$
for $x \ge 2$.