`int_(-pi//4)^(pi//4) \ (e^x sec^2 \ dx)/(e^(2x)-1)` is equal to (i)`0` (ii)`2` (iii)`e` (iv)none of these

Options

A. 0
B. 2
C. e
D. 2e

Explanation

To solve the integral

\[ I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{e^x \sec^2 x}{e^{2x} - 1} \, dx, \]

we will check if the integrand is an odd function. An odd function satisfies the property \( f(-x) = -f(x) \).

### Step 1: Define the function
Let

\[ f(x) = \frac{e^x \sec^2 x}{e^{2x} - 1}. \]

### Step 2: Calculate \( f(-x) \)
Now, we compute \( f(-x) \):

\[ f(-x) = \frac{e^{-x} \sec^2(-x)}{e^{-2x} - 1}. \]

Since \( \sec(-x) = \sec(x) \), we have \( \sec^2(-x) = \sec^2(x) \). Thus,

\[ f(-x) = \frac{e^{-x} \sec^2 x}{e^{-2x} - 1}. \]

### Step 3: Simplify \( f(-x) \)
Now, we rewrite \( e^{-2x} \) as \( \frac{1}{e^{2x}} \):

\[ f(-x) = \frac{e^{-x} \sec^2 x}{\frac{1}{e^{2x}} - 1} = \frac{e^{-x} \sec^2 x \cdot e^{2x}}{1 - e^{2x}} = \frac{e^{x} \sec^2 x}{1 - e^{2x}}. \]

### Step 4: Factor out -1
Notice that

\[ 1 - e^{2x} = -(e^{2x} - 1). \]

Thus, we can write:

\[ f(-x) = -\frac{e^{x} \sec^2 x}{e^{2x} - 1} = -f(x). \]

### Step 5: Conclusion about the function
Since \( f(-x) = -f(x) \), \( f(x) \) is an odd function.

### Step 6: Evaluate the integral
The integral of an odd function over a symmetric interval around zero is zero:

\[ I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} f(x) \, dx = 0. \]

### Final Answer
Thus, the value of the integral is

\[ \boxed{0}. \]