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This question is an exact duplicate of:

We are asked to prove the following integral;

$\int_0^x(\int_0^tf(u)du)dt=\int_0^xf(u)(x-u)dx$

We have to use integration by parts and then by this apply the fundamental theorem of calculus which I believe need to be applied to the derivative part of the one done by parts. I haven't seen anything like this before.

Any help would be greatly appreciated

Thanks in advance!

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marked as duplicate by RRL calculus Apr 30 at 7:06

This question was marked as an exact duplicate of an existing question.

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Your region is bound by $u=t$, $u = 0$, and $t = x$

Plot this out.

Just because you are working with u and t axes instead of x and y that shouldn't freak you out.

enter image description here

Swap the order of integration

$\int_0^x\int_u^x f(u)\, dt\, du$

And integrate with respect to $t$.

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