Let f be a continuous function satisfying the equation `int_(0)^(x)f(t)dt+int_(0)^(x)tf(x-t)dt=e^(-x)-1`, then find the value of `e^(9)f(9)` is equal to…………………..

Answer

8

Explanation

`int_(0)^(x)f(t)dt+int_(0)^(x)tf(x-t)dt=e^(-x)-1`
`rArr int_(0)^(x) f(t)dt+int_(0)^(x)(x-t)f(t)dt=e^(-x)-1`
`rArr int_(0)^(x)f(t)dt +int_(0)^(x)tf(t)dt=e^(-x)-1`
Differentiate w.r.t. x, we get
`f(x) + xf(x) + int_(0)^(x)f(t)dt-xf(x)=-e^(-x)`...........(1)
`rArr f(x) + int_(0)^(x)f(t)dt=-e^(-x)`
Again differentiate w.r.t x., we get
`rArr f^(')(x) + f(x)=e^(-x)`
`rArr e^(x)f^(')(x)+e^(x)f(x)=1`
`rArr (e^(x)f(x))^(')=1`
`rArr e^(x)f(x)=x+c`
From (1), `f(0)=-1, therefore c=-1`
`therefore f(x)e^(x)=x-1`
`therefore e^(9)f(9)=8`