Derivation of a distribution

Question

We consider the distribution

$$T(\phi)=\int_{\mathbb{R}}\phi(x)|\cos(x)|dx\quad\quad\forall \phi\in D(\mathbb{R})$$

Could you please help me to find the expressions of the derivatives $T^{(k)}$ for all $k\ge 1$?

Thanks in advance

Answer 1

Well, derivatives of distributions are defined by duality.

$T'(\phi)=(T', \phi) = (-1)\cdot (T, \phi') = (-1) \int_\mathbb{R} \phi'(x) \vert \cos(x)\vert dx$.

Try to use integration by parts now!