The value of $$\frac{e^{-\frac{\pi}{4}} + \int_0^{\frac{\pi}{4}} e^{-x}\tan^{50}x \ dx}{\int_0^{\frac{\pi}{4}} e^{-x}(\tan^{49}x + \tan^{51}x) \ dx}$$

$$I = \frac{e^{-\frac{\pi}{4}} + \int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{50}x \, dx}{\int_{0}^{\frac{\pi}{4}} e^{-x}\left(\tan^{49}x + \tan^{51}x\right) dx}$$

$$J_1 = \int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{50}x \, dx$$

$$J_2 = \int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{49}x(1 + \tan^2 x) \, dx = \int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{49}x\sec^2 x \, dx$$

$$\int \tan^{49}x\sec^2 x \, dx = \frac{\tan^{50}x}{50}$$

$$J_2 = \left[ e^{-x} \frac{\tan^{50}x}{50} \right]_{0}^{\frac{\pi}{4}} - \int_{0}^{\frac{\pi}{4}} \left(-e^{-x}\right) \frac{\tan^{50}x}{50} \, dx$$

$$J_2 = \frac{e^{-\frac{\pi}{4}}\tan^{50}\left(\frac{\pi}{4}\right)}{50} - 0 + \frac{1}{50}\int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{50}x \, dx$$

$$J_2 = \frac{e^{-\frac{\pi}{4}}}{50} + \frac{1}{50}\int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{50}x \, dx$$

$$50 J_2 = e^{-\frac{\pi}{4}} + \int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{50}x \, dx$$

$$I = \frac{50 J_2}{J_2} = 50$$