Let n be a positive integer. Find the continuous functions f:ℝ->ℝ with the property that integral from 1 to x of f(ln(t)) dt=x^n ln(x) for all positive real numbers x.

Differentiating the integral equation with respect to x we obtain: f(ln(x))=nx^n-1ln(x)+x^n-1=x^n-1=x^n-1(nln(x)+1), for any positive real number x.Let ln(x)=t, thus x=e^tand f(t)=e^t(n-1)(nt+1), for any t>0