Let y=y(x) be the solution of the differential equation d y ~d x + 2 x (1+x^2 )^2 y=x e ^ 1 (1+x^2 ) ; y(0)=0 . Then the area enclosed by the curve f(x)=y(x) e ^ - 1 (1+x^2 ) and the line y-x=4 is__________

Step-by-step solution

aligned & I F=e^ 2 x (1+x^2 )^2 d x =e^ -1 1+x^2 & y e^ -1 1+x^2 = x e^ 1 1+x^2 e^ -1 1+x^2 d x & y e^ -1 1+x^2 = x^2 2 +c & (0,0) C=0 & y(x)= x^2 2 e^ 1 1+x^2 & f(x)= x^2 2 aligned A= _ -2 ^4(x+4)- x^2 2 d x=18