If I = x+ ^3 x 2 x ~d x= P x+ Q | 2 x-1 2 x+1 |+ c , (where c is a constant of integration), then the values of P and Q are respectively

Step-by-step solution

aligned I & = x+ ^3 x 2 x d x & = x (1+ ^2 x ) 2 ^2 x-1 d x & = x (2- ^2 x ) 2 ^2 x-1 d x aligned aligned & Put x= t & - x ~d x= dt & I = t ^2-2 2 t ^2-1 dt & = 1 2 2 t ^2-4 2 t ^2-1 dt & = 1 2 [ 2 t ^2-1 2 t ^2-1 dt - 3 2 t ^2-1 dt ] & I= 1 2 t- 1 2 3 2 2 | 2 t-1 2 t+1 |+c & = 1 2 x- 3 4 2 | 2 x-1 2 x+1 |+ c & P = 1 2 and Q = -3 4 2 & aligned