Deducing From The Integral

$\textbf{Partially differentiating the equation wrt. x,}$ $\dfrac{\partial}{\partial x} \int_{1}^{xy} f(t) \, dt = \dfrac{\partial}{\partial x}(y\int_{1}^{x} f(t) \, dt + x\int_{1}^{y} f(t) \, dt)$

$yf(xy) = y f(x) + \int_{1}^{y} f(t) dt$

$\textbf{Put x=1,}$

$yf(y) = 3y + \int_{1}^{y} f(t) dt$

$\textbf{Differentiate wrt. y,}$

$y f'(y) + f(y) = 3 + f(y)$

$f'(y) = \frac{3}{y}$

$\textbf{Integrating,}$

$f(y) = 3 logy + c$

$\textbf{Use the fact that f(1) = 3 to find c , so-}$

$f(e) = 6$