Calculus and Patterns II

Consider $\displaystyle I_n= \int_{0}^{\frac{\pi}{2}}(\ln\sin{x})^ndx$ , where $n\in\mathbb{N}$ .

$\int_{0}^{\frac{\pi}{2}}\ln\sin{x}dx \approx -1.0879 \\ \int_{0}^{\frac{\pi}{2}}(\ln\sin{x})^2dx \approx 2.0466 \\ \int_{0}^{\frac{\pi}{2}}(\ln\sin{x})^3dx \approx -6.0418 \\ \int_{0}^{\frac{\pi}{2}}(\ln\sin{x})^4dx \approx 24.0529$

As you can see, $\left \lfloor \left| {I_n}\right| \right \rfloor$ is always equal to or close to $n!$ . For what first value of $n$ , $\left \lfloor \left| {I_n}\right| \right \rfloor$ stops being equal to $n!$ ?

Bonus: Can you explain why this pattern follows?