Question

求帮忙做一道高数题

Answer 1

如图所示：

追答

抱歉，这个才对。

Answer 2

let
u= x-t
du =-dt
t=0, u=x
t=x , u=0
∫(0->x) f(x-t) dt
=∫(x->0) f(u) (-du)
=∫(0->x) f(t) dt
f(x)
= x+∫(0->x) tf'(x-t) dt

= x-∫(0->x) tdf(x-t)
=x -[ tf(x-t) ]|(0->x) + ∫(0->x) f(x-t) dt
=x - xf(0) +∫(0->x) f(x-t) dt
=x - xf(0) +∫(0->x) f(t) dt (1)
两边求导
f'(x) = 1-f(0) + f(x)
f'(x) - f(x) = 1-f(0)
let
f(x)= Ae^(x) -[ 1-f(0) ]
from (1)
f(x) =x - xf(0) +∫(0->x) f(t) dt
f(0) = 0
f(x)= Ae^(x) -[ 1-f(0) ]
0= A -1
A= 1
ie
f(x) =e^(x) - 1