`int_(-pi//4)^(pi//4)(e^(x)(x^(3)sinx))/(e^(2x)-1)dx` equals

Options

A. 0
B. 1
C. `-1`
D. e

Answer

To solve the integral

\[ I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{e^x x^3 \sin x}{e^{2x} - 1} \, dx, \]

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

### Step 1: Define the function

Let

\[ f(x) = \frac{e^x x^3 \sin x}{e^{2x} - 1}. \]

### Step 2: Evaluate \( f(-x) \)

Now we calculate \( f(-x) \):

\[ f(-x) = \frac{e^{-x} (-x)^3 \sin(-x)}{e^{2(-x)} - 1}. \]

### Step 3: Simplify \( f(-x) \)

Using the properties of exponents and trigonometric functions, we have:

\[ f(-x) = \frac{e^{-x} (-x^3)(-\sin x)}{e^{-2x} - 1} = \frac{e^{-x} x^3 \sin x}{e^{-2x} - 1}. \]

### Step 4: Rewrite the denominator

The denominator can be rewritten as:

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

Thus,

\[ f(-x) = \frac{e^{-x} x^3 \sin x}{-(e^{2x} - 1)} = -\frac{e^{-x} x^3 \sin x}{e^{2x} - 1}. \]

### Step 5: Relate \( f(-x) \) to \( f(x) \)

Now, we can express \( f(-x) \) in terms of \( f(x) \):

\[ f(-x) = -\frac{e^{-x} x^3 \sin x}{e^{2x} - 1}. \]

Notice that \( e^{-x} = \frac{1}{e^x} \), so we can write:

\[ f(-x) = -f(x). \]

### Step 6: Conclude that \( f(x) \) is an odd function

Since \( f(-x) = -f(x) \), we conclude that \( f(x) \) is an odd function.

### Step 7: Apply the property of definite integrals

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 Result

Thus, the value of the integral is:

\[ \boxed{0}. \]

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