Fourier transform and distibution beloning to S' I need prove, that a distribution $$\langle F_f,\phi \rangle= p.v. \int\limits_{-1/2}^{1/2}\frac{\phi(t)}{t\cdot \ln{|t|}}\mathrm{d}t$$ belongs to $S^\prime$ (adjoint to Schwartz space) and I need find, Fourier transform of the distribution.

If $\phi\in\mathcal{S}$, $\psi(t)=\frac{\phi(t)-\phi(0)}{t}\in C^\infty$, and by the Mean Value Theorem, $\|\psi\|_{L^\infty}\le\|\phi'\|_{L^\infty}$ Thus,
$$
\begin{align}
\lim_{\epsilon\to0}\left(\int_{-1/2}^{-\epsilon}\frac{\phi(t)}{t\log|t|}\,\mathrm{d}t+\int_{\epsilon}^{1/2}\frac{\phi(t)}{t\log|t|}\,\mathrm{d}t\right)
&=\lim_{\epsilon\to0}\left(\int_{-1/2}^{-\epsilon}\frac{\psi(t)}{\log|t|}\,\mathrm{d}t+\int_{\epsilon}^{1/2}\frac{\psi(t)}{\log|t|}\,\mathrm{d}t\right)\\
&=\int_{-1/2}^{1/2}\frac{\psi(t)}{\log|t|}\,\mathrm{d}t\tag{1}
\end{align}
$$
Therefore
$$
\begin{align}
\left|\text{p.v.}\int_{-1/2}^{1/2}\frac{\phi(t)}{t\log|t|}\,\mathrm{d}t\right|&=\left|\int_{-1/2}^{1/2}\frac{\psi(t)}{\log|t|}\,\mathrm{d}t\right|\\
&\le\left|\int_{-1/2}^{1/2}\frac{1}{\log|t|}\,\mathrm{d}t\right|\;\|\psi\|_{L^\infty}\\
&\le\frac{1}{\log(2)}\|\phi'\|_{L^\infty}\tag{2}
\end{align}
$$
Inequality $(2)$ says that $\langle F_f,\cdot\rangle\in\mathcal{S}'$.
The Fourier Transform of the distribution would be
$$
\begin{align}
\text{p.v.}\int_{-1/2}^{1/2}\frac{e^{-2\pi i\xi t}}{t\log|t|}\,\mathrm{d}t
&=-i\int_{-1/2}^{1/2}\frac{\sin(2\pi\xi t)}{t\log|t|}\,\mathrm{d}t\\
&=-2i\int_0^{1/2}\frac{\sin(2\pi\xi t)}{t\log|t|}\,\mathrm{d}t\tag{3}
\end{align}
$$
I can not find a closed form for $(3)$.
