Evaluate $\int_{0}^{\frac{\pi}{2}} \sqrt{1+\cos^2 x}\, dx$ [duplicate]

That is not an evaluable integral. In terms of elementary functions I mean, unless you consider Elliptic integrals as elementary.

First of all, we cannot even try any numerical or Series evaluation since the integral is unbounded. Your way of substitution is good, because as other users have already pointed out, the result is in terms of Special functions called the Elliptic Functions. That is:

$$\int\sqrt{1 + \cos^2(x)}\ \text{d}x = \sqrt{2}\ \text{E}\left(x, \frac{1}{2}\right)$$

Where $E$ stands for the elliptic integral of Second Kind.

More here: https://en.wikipedia.org/wiki/Elliptic_integral