merator and denominator have the same form. They both involve an integral of $e^{-x} \tan^n x$ from 0 to $\pi / 4$, where $n$ is an integer. Let's denote this integral as $I(n)$:

$$I(n) = \int_0^{\pi / 4} e^{-x} \tan^n x dx$$

We can then rewrite the original expression in terms of $I(n)$:

$$\frac{e^{-\pi/4} + I(50)}{I(49) + I(51)}$$

Now, we'll apply the method of integration by parts, which states that for two functions $u(x)$ and $v(x)$:

$$\int u dv = uv - \int v du$$

We'll choose:

$$u = \tan^n x, \quad dv = e^{-x} dx$$

Then we get:

$$du = n \tan^{n-1} x \sec^2 x dx, \quad v = -e^{-x}$$

Applying integration by parts, we have:

$$I(n) = -e^{-x} \tan^n x \Bigg|_0^{\pi / 4} + n \int_0^{\pi / 4} e^{-x} \tan^{n-1} x \sec^2 x dx$$

Since $\tan(\pi / 4) = 1$, the first term evaluates to:

$$-e^{-\pi / 4}$$

The second term becomes:

$$n \int_0^{\pi / 4} e^{-x} \tan^{n-1} x (1 + \tan^2 x) dx = n \int_0^{\pi / 4} e^{-x} \tan^{n-1} x dx + n \int_0^{\pi / 4} e^{-x} \tan^{n+1} x dx$$

This is equal to:

$$n(I(n-1) + I(n+1))$$

So we have:

$$I(n) = -e^{-\pi / 4} + n(I(n-1) + I(n+1))$$

Now we can substitute $n = 50$ into this equation:

$$I(50) = -e^{-\pi / 4} + 50(I(49) + I(51))$$

So the original expression becomes:

$$\frac{e^{-\pi/4} + I(50)}{I(49) + I(51)} = \frac{e^{-\pi/4} - e^{-\pi / 4} + 50(I(49) + I(51))}{I(49) + I(51)} = 50$$

Definite Integration

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