If I = e ^x e ^ 4 x + e ^ 2 x +1 ~d x and J = e ^ -x e ^ -4 x + e ^ -2 x +1 ~d x then for any arbitrary constant c, the value of J - I equals

Step-by-step solution

aligned J - I & = ( e ^ -x e ^ -4 x + e ^ -2 x +1 - e ^x e ^ 4 x + e ^ 2 x +1 ) d x & = ( e ^ 3 x e ^ 4 x + e ^ 2 x +1 - e ^x e ^ 4 x + e ^ 2 x +1 ) d x & = ( e ^ 2 x -1 ) e ^x e ^ 4 x + e ^ 2 x +1 ~d x aligned Put e ^x= t e ^x ~d x= dt J - I = t ^2-1 t ^4+ t ^2+1 dt = 1- 1 t ^2 ( t + 1 t )^2-1 dt Put t + 1 t =y (1- 1 t ^2 ) dt = d y J - I = d y y^2-1^2 = 1 2 | y-1 y+1 |+ c aligned & = 1 2 | t + 1 t -1 t + 1 t +1 |+ c & = 1 2 | t ^2- t +1 t ^2+ t +1 |+ c & = 1 2 | e ^ 2 x - e ^x+1 e ^ 2 x + e ^x+1 |+ c aligned