If $\int_{1/2}^2\frac{1}{x}cosec^{101}(x-\frac{1}{x})dx=k$ then the value of k is:

0

$I=\int\limits_{1/2}^2\frac{1}{x}.cosec^{101}(x-\frac{1}{x})dx=\int\limits_{2}^{1/2}t\,cosec^{101}(\frac{1}{t}-t)(-\frac{dt}{t^2})$  $[x=\frac{1}{t}⇒dx=-\frac{1}{t^2}dt]$

$⇒I=\int\limits_{2}^{1/2}\frac{1}{t}.cosec^{101}(t-\frac{1}{t})dt=-\int\limits_{1/2}^{2}\frac{1}{x}cosec^{101}(x-\frac{1}{x})dx=-I⇒I=0$