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aytcH (pinky@NOSPAM.for-president.com) wants to prove:
: Integration by Parts: integral of udv = uv - integral of vdu

: I think I find their derivatives are the same, then the formula is proven.
<details invoving a double integral omitted.>

You have the exact RIGHT idea.  Take the derivatives with respect to x.
The chain rule is required to differentiate the integrals.
(On the left:  differentiate with respect to v and multiply by dv/dx).

WARNING:  Two functions with the SAME derivative do not have to be
equal, they can differ by a constant.  But the indefinite integrals
in your formula are only defined up to a constant (they include an
undetermined constant).  So this formula IS proven just by showing
that the derivatives are equal.

If you forget about that hidden constant, you can use the formula
to "prove" that 1=0!
Consider u=1/x and v=x  (so du=dx and dv=-(1/x^2)dx):
  Integral of (1/x)dx = 1 - Integral x*(-1/x^2)dx = 1 + Integral (1/x)dx
Cancel the Integral on both sides to get:
  0 = 1.

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