Dirac Delta Integration by Parts

Question

$$\int_{-\infty}^{\infty}\left(\sin ^{2} x+2 \tan x\right) \delta^{\prime}(x)$$

$1.$ Is this integral missing a $dx$ at the end?

$2.$ Assuming that it is then performing integration by parts leaves me with $$ \left.\left[(\sin ^{2}(x)+2 \tan (x)\right) \delta(x)\right]_{-\infty}^{\infty}-\int_{-\infty}^{\infty} 2[\cos (x)\sin(x)+\sec^2(x)] \delta(x) d x $$

But how does one evaluate the left hand side?

Answer 1

1. Technically yes.
2. The first term is just zero; in fact this always happens when $f$ is a test function and $d$ is a distribution*, so that distributional integration by parts reads $\int f d' = -\int f' d$ (in shorthand). The second term is just evaluated using the definition of $\delta$.

* Here I mean either "$f$ is compactly supported smooth and $d$ is a distribution" or "$f$ is Schwarz class and $d$ is a tempered distribution".

Answer 2

When we define the distributional derivative, the form is like \begin{align*} \left<\partial^{\alpha}\varphi,f\right>=(-1)^{|\alpha|}\left<\varphi,\partial^{\alpha}f\right> \end{align*} for good functions $f$, we define in this way such that the integration by parts is allowed.