The limit is $$\lim_{n\rightarrow\infty} \int_0^{\frac{\pi}{2}} \frac{e^x}{\left(\left(\frac{2}{\pi}\right) x\right)^n + 1} \cdot \left| \sin(x) \sin(nx) \sin(2nx) \right| \, dx=\frac{2 \left(1+e^{\pi /2}\right)}{3 \pi }=1.23302\cdots.$$$$\lim_{n\rightarrow\infty} \int_0^{\pi/2} \frac{e^x}{\left(2x/\pi\right)^n + 1} \left| \sin(x) \sin(nx) \sin(2nx) \right| \, dx=\frac{2 \left(1+e^{\pi /2}\right)}{3 \pi }=1.23302\cdots.$$
For large $n$ and $0<x<\pi/2$ the factor $|\sin(nx)\sin(2nx)|$ may be replaced by its average $\frac{4}{3\pi}$, while the denominator $\left(2x/\pi\right)^n + 1$ may be replaced by unity, resulting in $$\lim_{n\rightarrow\infty}I(n)=\frac{4}{3\pi}\int_0^{\pi/2} e^x \sin x \, dx=\frac{2 \left(1+e^{\pi /2}\right)}{3 \pi }.$$