The value of e^-( )/(4)+ _0^( )/(4)e^-x ^50xdx _0^( )/(4)e^-x( ^49x+ ^51x)dx is

Question

The value of $\frac{{e}^{-\frac{\pi }{4}}+\int _{0}^{\frac{\pi }{4}}{e}^{-x}{	an }^{50}xdx}{\int _{0}^{\frac{\pi }{4}}{e}^{-x}({	an }^{49}x+{	an }^{51}x)dx}$ is

Accepted Answer

We're given the expression:

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

Notice that the integrals in the numerator 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:

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

Solutions · 1