If `int_(1/2)^2 1/x cosec^(101)(x-1/x)dx=k` then the value of k is :

Options

A. 1
B. `1//2`
C. 0
D. `1//101`

Answer

To solve the integral \( k = \int_{\frac{1}{2}}^{2} \frac{1}{x} \csc^{101}\left(x - \frac{1}{x}\right) \, dx \), we will use a substitution and properties of integrals.

### Step 1: Substitution
Let \( x = \frac{1}{t} \). Then, we differentiate to find \( dx \):
\[ dx = -\frac{1}{t^2} dt \]

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

### Step 3: Substitute in the integral
Now substituting \( x = \frac{1}{t} \) into the integral:
\[ k = \int_{2}^{\frac{1}{2}} \frac{1}{\frac{1}{t}} \csc^{101}\left(\frac{1}{t} - t\right) \left(-\frac{1}{t^2}\right) dt \]
This simplifies to:
\[ k = \int_{2}^{\frac{1}{2}} -\frac{t}{1} \csc^{101}\left(\frac{1}{t} - t\right) dt \]
Reversing the limits gives:
\[ k = \int_{\frac{1}{2}}^{2} \frac{1}{t} \csc^{101}\left(t - \frac{1}{t}\right) dt \]

### Step 4: Recognizing symmetry
Notice that \( \csc(-x) = -\csc(x) \). Therefore:
\[ \csc^{101}\left(t - \frac{1}{t}\right) = -\csc^{101}\left(-\left(t - \frac{1}{t}\right)\right) \]
This implies that the integrand is odd with respect to the transformation \( x \rightarrow \frac{1}{x} \).

### Step 5: Adding the two integrals
Now we have two expressions for \( k \):
1. \( k = \int_{\frac{1}{2}}^{2} \frac{1}{x} \csc^{101}\left(x - \frac{1}{x}\right) dx \)
2. \( k = \int_{\frac{1}{2}}^{2} \frac{1}{t} \csc^{101}\left(t - \frac{1}{t}\right) dt \)

Adding these two equations:
\[ 2k = \int_{\frac{1}{2}}^{2} \left( \frac{1}{x} \csc^{101}\left(x - \frac{1}{x}\right) + \frac{1}{x} \csc^{101}\left(-\left(x - \frac{1}{x}\right)\right) \right) dx \]
Since the integrand sums to zero:
\[ 2k = 0 \implies k = 0 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{0} \]