[Integral symmetries] Cauchy Riemann theorem like analogues in reals:

$$\frac{d^n}{dx^n}f=\lambda\int^{(m)}fdx$$

(where $f=\sin, \cos,e^x$ (possibly more?),$\lambda \in \mathbb{R}$)

For example

$$\int x^n \sin x e^x dx$$
Consider

$(uv)''=u''v+2u'v'+uv''$

Plug $u=\sin x$, $v=e^x$. Then

$(uv)''=-uv+2u'v+uv$

Note how the antisymmetry of $\sin x$ wrt the functional $()''$ results in cancellation. Hence

$(uv)''=2u'v$

Now it is easy to check that $\cos x$ is also antisymmetric wrt $()''$ thus we could have start with $u' =\sin x\implies u = -\cos x$. Hence