Question

If `f` is a continuous function such that `int_0^xf(t)dt=xe^(2x)` + `int_0^xe^-tf(t)dt` for all `x`, find an explicit formula for `f(x)` ?

If `f` is a continuous function such that `int_0^xf(t)dt=xe^(2x)` + `int_0^xe^-tf(t)dt` for all `x`, find an explicit formula for `f(x)` ?

Answer 1

`f(x)=(e^(3x)(2x+1))/(e^x-1)`

Explanation:

`int_0^xf(t)dt=xe^(2x)+int_0^xe^(-t)f(t)dt`

Differentiate both sides with respect to `x` (hint: use the fundamental rule of calculus and the product rule):

`f(x)=2xe^(2x)+e^(2x)+e^(-x)f(x)`

Now, arrange both sides to get
`f(x)=(e^(2x)(2x+1))/(1-e^(-x))`
`color(white)(f(x))=(e^(3x)(2x+1))/(e^x-1)`