`int_(1//2)^(2) (1)/(x) cosec^(101) (x-(1)/(x))dx`=

Options

A. `1//4`
B. 1
C. 0
D. none

Answer

To solve the integral

\[ I = \int_{\frac{1}{2}}^{2} \frac{1}{x} \csc^{101} \left( x - \frac{1}{x} \right) dx, \]

we will use a substitution method.

### Step 1: Substitution
Let \( t = \frac{1}{x} \). Then, we differentiate \( t \) with respect to \( x \):

\[ \frac{dt}{dx} = -\frac{1}{x^2} \implies dx = -\frac{1}{t^2} dt. \]

### Step 2: Change the limits of integration
When \( x = \frac{1}{2} \), \( t = 2 \). When \( x = 2 \), \( t = \frac{1}{2} \). Thus, the limits change from \( x: \frac{1}{2} \to 2 \) to \( t: 2 \to \frac{1}{2} \).

### Step 3: Substitute in the integral
Now we substitute \( x \) and \( dx \) in the integral:

\[ I = \int_{2}^{\frac{1}{2}} \frac{1}{\frac{1}{t}} \csc^{101} \left( \frac{1}{t} - t \right) \left(-\frac{1}{t^2}\right) dt. \]

This simplifies to:

\[ I = \int_{2}^{\frac{1}{2}} -t \csc^{101} \left( \frac{1}{t} - t \right) \frac{1}{t^2} dt = \int_{2}^{\frac{1}{2}} -\csc^{101} \left( \frac{1}{t} - t \right) dt. \]

### Step 4: Change the order of integration
Reversing the limits of integration gives us:

\[ I = \int_{\frac{1}{2}}^{2} \csc^{101} \left( \frac{1}{t} - t \right) dt. \]

### Step 5: Relate the integrals
Now we have:

\[ I = \int_{\frac{1}{2}}^{2} \csc^{101} \left( \frac{1}{x} - x \right) dx. \]

### Step 6: Combine the integrals
Adding the two expressions for \( I \):

\[ 2I = \int_{\frac{1}{2}}^{2} \left( \csc^{101} \left( x - \frac{1}{x} \right) + \csc^{101} \left( \frac{1}{x} - x \right) \right) dx. \]

### Step 7: Use the property of cosecant
Using the property that \( \csc(-\theta) = -\csc(\theta) \), we find that:

\[ \csc^{101} \left( \frac{1}{x} - x \right) = -\csc^{101} \left( x - \frac{1}{x} \right). \]

Thus, the integral simplifies to:

\[ 2I = \int_{\frac{1}{2}}^{2} 0 \, dx = 0. \]

### Step 8: Solve for \( I \)
This implies:

\[ I = 0. \]

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

\[ \boxed{0}. \]