TITLE: Difficult Definite Integral $\int_{0}^{\frac{\pi}{2}} \sqrt{1+2\cos^2\left(\frac{\pi}{2} - x\right)} + \sin x\, dx$
QUESTION [12 upvotes]: I have spent several days trying to solve this integral, but to no avail. This isn't from a textbook, but a challenge problem given to me by a professor. I am not looking for anyone to give me the solution, but just to lead me in the right direction.
The problem is to compute the following integral:
\begin{equation}
\int_{0}^{\frac{\pi}{2}} \sqrt{1+2\cos^2\left(\frac{\pi}{2} - x\right)} + \sin x\, dx
\end{equation}
When first approaching this problem I tried to utilize the cofunction identity:
\begin{equation}
\cos\left(\frac{\pi}{2}-x\right) = \sin x
\end{equation}
The integral then became:
\begin{equation}
\int_{0}^{\frac{\pi}{2}} \sqrt{1+2\sin^2x} + \sin x\, dx
\end{equation}
I have tried several things from this point such as using the formulas 
\begin{equation}
\sin^2x = \frac{1}{2}[1-\cos(2x)]
\end{equation}
The integral then became:
\begin{equation}
\int_{0}^{\frac{\pi}{2}} \sqrt{2-\cos(2x)} + \sin x\, dx
\end{equation}
The issue is I have tried several run arounds(of which I will not post each) with identities and other methods, but I seem to be hitting dead ends. Also, I want to mention that I'm trying to solve this using elementary methods only. I only have experience up to calculus II. Any constructive criticism or comments would be greatly appreciated! Thank you.

REPLY [6 votes]: As said in comments and answers, you are facing elliptic integrals that you cannot evaluate easily.
$$\int_0^{\frac \pi 2}\sqrt{1+k \sin ^2(x)}\,dx=E(-k)$$ where appears the complete elliptic integral of the second kind.
However, we can build quite good approximations. I give you one I produced years ago (for rather small values of $k$)  using Padé approximants built at $k=0$.
$$E(-k) \simeq \frac \pi 2 \,\frac{1+\frac{39575 }{28464}k+\frac{20621} {37952}k^2+\frac{129235}{2428928}k^3 } {1+\frac{32459}{28464}k+\frac{34741 }{113856}k^2+\frac{79037
  }{7286784}k ^3 }$$ which is quite good for the range $0\leq k \leq 4$.
Using $k=-2$, we should get , as an approximation, $\frac{5810969}{8357946}\pi\approx 2.18423$ while the exact value would be $E(-2)\approx 2.18444$.
Edit
The approximation I wrote was made more than fourty years ago and it was, at that time, a hard work. Just for the fun of it, I made, after answering, a better one which took me a few minutes .... thanks to a CAS. It is
$$E(-k) \simeq \frac \pi 2 \,\frac{1+\frac{133542997 }{70902928}k+\frac{1325913585 }{1134446848}k^2+\frac{1210596065
  }{4537787392}k^3+\frac{4808786003 }{290418393088} k^4} {1+\frac{115817265 }{70902928}k+\frac{915821721 }{1134446848}k^2+\frac{553597479
  }{4537787392}k^3+\frac{777708891 }{290418393088}k^4 }$$
Using $k=-2$, we should get , as an approximation, $\frac{214931493555  }{309110015222}\pi\approx 2.184424$ while the exact value would be $E(-2)\approx 2.184438$.