arc length question calculus

Question

we have to calculate the arc length of the following function $y=\sqrt{(\cos2x)} dx$ in the interval $[0 ,\pi/4]$. I know the arc length formula but following it becomes an integral thats really complex...need help....

Answer 1

$\int_{0}^{\frac{\pi}{4}} \sqrt{\cos2x} dx = \frac{1}{2}\int_{0}^{\frac{\pi}{4}} \sqrt{\cos{u}} du = \frac{1}{2}\cdot \:2\text{E}\left(\frac{u}{2}|\:2\right) = \frac{1}{2}\cdot \:2\text{E}\left(\frac{2x}{2}|\:2\right) = \text{E}\left(x|2\right)+C$, where $\text{E}\left(x|m\right)$ is the elliptic integral of the second kind. See: https://math.stackexchange.com/a/19786/733593

Answer 2

I got the following integral $$\int_0^{\frac{\pi }{4}} \sqrt{\sin (2 x) \tan (2 x)+1} \, dx$$ no hope that this has an algebraic solution.