If the integral $525 \int_0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\cos ^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x$ is equal to $(n \sqrt{2}-64)$, then $n$ is equal to

Let $\mathrm{f}:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$, If the $\lim _{x \rightarrow 0} \frac{x \int_0^x f(t) d t}{e^{x^2}-1}=\alpha$, then $...