Let f:[0, ) R be differentiable function such that f( x )=1-2 x + ₀^x e^ x-t f(t) dt for all x [0, ) . Then the area of the region bounded by y =f( x ) and the coordinate axes is

Step-by-step solution

aligned & y=1-2 x+e^x ₀^x e^ -t f(t) d t & d y d x =-2+e^ -x e^x f(x)+e^x ₀^x e^ -t f(t) d t & d y d x =-2+y+y+2 x-1 & d y d x -2 y=(2 x-3) & y e^ -2 x = (2 x-3) d x e^ -2 x & y e^ -2 x = -(2 x-3) 2 e^ -2 x + e^ -2 x d x & y e^ -2 x = -(2 x-3) 2 e^ -2 x - 1 2 e^ -2 x +c & f(0)=1 c=1- 3 2 + 1 2 =0 aligned aligned & y=- (2 x-3) 2 - 1 2 & y=-x+1 & x+y=1 & area = 1 2 (1)(1)= 1 2 aligned