Evaluating $\lim \limits_{n\to \infty}\,\,\, n\!\! \int\limits_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx$ 
Evaluate the following limit:
  $$\lim \limits_{n\to \infty}\,\,\, n\!\! \int\limits_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$

I have done the problem .
My method:
First I applied L'Hôpital's rule as it can be made of the form $\frac0 0$. Then I used weighted mean value theorem and using sandwich theorem reduced the limit to an integral which could be evaluated using properties of define integration .
I would like to see other different ways to solve for the limit.

You can have a close form solution; infact if $Re(1/n)>-1$ you have that the integral collapse in:
$$\int_{0}^{\pi/2}\left[1-(\sin(x))^{1/n}\right]dx=\frac{1}{2} \left(\pi -\frac{2 \sqrt{\pi } n \Gamma
   \left(\frac{n+1}{2 n}\right)}{\Gamma
   \left(\frac{1}{2 n}\right)}\right)$$
So we define:
$$y(n)=\frac{n}{2} \left(\pi -\frac{2 \sqrt{\pi } n \Gamma
   \left(\frac{n+1}{2 n}\right)}{\Gamma
   \left(\frac{1}{2 n}\right)}\right)$$
And performing the limit:
$$\lim_{n \rightarrow + \infty}y(n)=-\frac{1}{4} \pi  \left[\gamma +\psi
   ^{(0)}\left(\frac{1}{2}\right)\right]$$
