Find $\lim_{n\to \infty}\int _0^{\frac{\pi}{2}} \sqrt{1+\sin^nx}$

Define

$$I_n=\int _0^{\frac{\pi}{2}} \sqrt{1+\sin^nx}\, dx$$

I have to show this sequence is convergent and find its limit. I proved it is decreasing: $\sin^{n+1} x \le \sin^n x \implies I_{n+1} \le I_n$. Also, it is bounded because:

$$0 \le \sin^n x \le 1 \implies \frac\pi{2} \le \int _0^{\frac{\pi}{2}} \sqrt{1+\sin^n x}\, dx\le \frac{\pi\sqrt{2}}{2}$$

so it is convergent. I'm stuck at finding the limit. I think it should $\frac{\pi}{2}$ but I'm not sure.

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