Finding a function from a definite integral I have to find the explicit expression of the function $f$, knowing that:
$$\int_{0}^{x} f(t) \,dt = xe^{2x}+\int_{0}^{x} e^{-t}f(t) \,dt$$
I've recognized the integration by parts pattern in the right side of the equation, but I'm not able to move forward. Can you help me? Thanks in advance!
 A: My first instinct when seeing a function under an integral is to differentiate it outside. Thus, taking derivatives with respect to $x$ on both sides of the equation, we get:
$$
f(x) = e^{2x} + 2xe^{2x} + e^{-x} f(x)
$$
Then, by rearranging and solving for $f(x)$, we get:
$$
f(x) = \frac{e^{2x}(1 + 2x)}{1 - e^{-x}}
$$
Strictly speaking, one should also worry about the convergence of these integrals since there is a discontinuity at zero, but that may be outside the scope of this problem, and informally speaking those discontinuities can be seen to "cancel" out on each side of the equation.
A: Taking a derivative of both sides of the equation gives
$$ f(x) =  \frac{(1 + 2 x)e^{2x}}{1 - e^{-x}}$$
which should be the answer.
A: As it was pointed out in other answers, by differentiating the original equality, we conclude that the only expression that could work is
$$
f(x)=\dfrac{(2x+1)e^{2x}}{1-e^{-x}}.
$$
However, this candidate solution must be verified and, as it turns out, it leads to divergent integrals in the original equality. The final conclusion is that there exists no function $f$ in such conditions (at least not when the regularity of $f$ allows differentiation of the equality).
