y' = \dfrac{1}{\sec x} \cdot \sec x \tan x = \tan x. 1 + (y')^{2} = 1 + \tan^{2} x = \sec^{2} x.

\displaystyle L = \int_{0}^{\frac{\pi}{4}} \sqrt{\sec^{2} x} dx = \int_{0}^{\frac{\pi}{4}} \sec x dx

\displaystyle L = [\ln|\sec x + \tan x|]_{0}^{\frac{\pi}{4}} = \ln(\sqrt{2} + 1) - \ln(1 + 0) = \ln(\sqrt{2} + 1)