DeltaFunctionPrime[1 - x]
is the derivative of the Dirac delta-function .
Overview, Convolute, DeltaFunction, DeltaFunctionDoublePrime, Integrate2, SimplifyDeltaFunction.
[1 - x] DeltaFunctionPrime
[DeltaFunctionPrime[1 - x] f[x], {x, 0, 1}] Integrate2
[DeltaFunctionPrime[1 - x] x^2, {x, 0, 1}] Integrate2