₀^ x x x cosec x d x is equals to

Step-by-step solution

Let I= ₀^ x x x cosec x d x aligned Using property I= & = ₀^ f(x) d x & = ₀^ f(0+ -x) d x aligned Then, aligned I & = ₀^ ( -x) ( -x) ( -x) cosec ( -x) d x I & = ₀^ ( -x) x x cosec x d x aligned On adding Eqs. (i) and (ii), we get aligned & 2 I= ₀^ x(x+ -x) x cosec x d x & 2 I= ₀^ x x cosec x d x & 2 I= ₀^ ^2 x d x & 2 I= ₀^ 1- 2 x 2 d x & 2 I= [ x 2 - 2 x 4 ]₀^ & 2 I= [ 2 -0 ] aligned I= ^2 4