prove equality - definite integral of definite integral

Question

I need help to prove this expression.
I can check that it is correct but can not prove it. Please help.

$$\int_0^x \int_0^t f(v) dv dt = \int_0^x (x-t) f(t) dt $$

How do you check that it is correct? – Oct 23 '15 at 00:09 — , Oct 23 '15 at 00:09

score 0 · Answer 1 · answered Oct 23 '15 at 00:24

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.

prove equality - definite integral of definite integral

1 Answers1