# prove equality - definite integral of definite integral

I need help to prove this expression.
$$\int_0^x \int_0^t f(v) dv dt = \int_0^x (x-t) f(t) dt$$
Integrate by parts: $$\int_0^x \int_0^t f(v) \, dv \, dt = \left[ (t-x) \int_0^t f(v) \, dv \right]_0^x - \int_0^x (t-x) f(t) \, dt,$$ by carefully choosing $t-x$ as an antiderivative of $1$. Then the term in brackets vanishes and you have the formula you wanted.