_ 1 2 ^2 1 x cosec ¹⁰¹ (x- 1 x ) d x=

Step-by-step solution

Consider the integral I = _ 1/2 ^2 1 x ¹⁰¹ (x - 1 x ) dx . Applying the substitution x = 1/t yields dx = -1/t^2 dt , with limits transforming as x = 1/2 t = 2 and x = 2 t = 1/2 . The integrand transforms as follows: 1/x t , and x - 1/x 1/t - t = -(t - 1/t) . Since (- ) = - and the exponent is odd, ¹⁰¹(-(t-1/t)) = - ¹⁰¹(t-1/t) . Substituting gives: I = ₂^ 1/2 t (- ¹⁰¹(t-1/t)) (-1/t^2) dt = ₂^ 1/2 1 t ¹⁰¹ (t- 1 t ) dt Reversing the limits introduces a sign change: I = - _ 1/2 ^2 1 t ¹⁰¹ (t- 1 t ) dt Since the variabl