Problem with integrating DiracDelta
I also vote for a bug. The easiest way to see it is using an undefined function,
Integrate[DiracDelta[Cos[x]] f[x], {x, 0, Pi}]
which gives f[Pi/2]
. So for an undefined function it gives the right answer but for f=Sin
it does not give Sin[Pi/2]
. Hence a Bug.
I'd consider it a bug.
Reason: I would consider following identity: $$\delta(f(x))=\sum_i\frac{\delta(x-x_i)}{|f'(x_i)|}$$
Where $\forall x_i : f(x_i)=0$
This leads us to: $$\delta(\cos x)=\frac{\delta(x-\pi/2)}{|-\sin(\pi/2)|}=\delta(x-\pi/2)$$
So we conclude:
$$\int_0^\pi\text{d}x\;\delta(x-\pi/2)\cdot\sin x=\sin(\pi/2)=1$$
The same for UnitStep
. And Mathematica realizes this:
Integrate[DiracDelta[x - Pi/2]*Sin[x], {x, 0, Pi}]
Integrate[DiracDelta[x - Pi/2]*UnitStep[x], {x, 0, Pi}]
1
1
With FullSimplify before:
FullSimplify[DiracDelta[x - Pi/2]*Sin[x], 0 < x < Pi]
FullSimplify[DiracDelta[x - Pi/2]*UnitStep[x], 0 < x < Pi]
2 DiracDelta[[Pi] - 2 x]
2 DiracDelta[[Pi] - 2 x]
Which evaluates in Integrate to:
Integrate[2 DiracDelta[\[Pi] - 2 x], {x, 0, Pi}]
1
So, my vote is on bug.