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

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

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  • $\begingroup$ How do you check that it is correct? $\endgroup$ – user99914 Oct 23 '15 at 0:09
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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.

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