= Solution
On either open half-line, part (i) says $\widehat u=-v'$. Differentiating the formula in part (iii) makes all elementary terms cancel except the <logarithm> and the bracket. Therefore
$$
\widehat u(\lambda)=-\log|\lambda|+f_+(\lambda),
\qquad \lambda>0,
$$
where
$$
f_+(\lambda)=
\int_\lambda^1\frac{e^{-ix}-1}{x}\,dx
+\int_1^\infty\frac{e^{-ix}}x\,dx,
$$
and
$$
\widehat u(\lambda)=-\log|\lambda|+f_-(\lambda),
\qquad \lambda<0,
$$
where
$$
f_-(\lambda)=
\int_{|\lambda|}^1\frac{e^{ix}-1}{x}\,dx
+\int_1^\infty\frac{e^{ix}}x\,dx.
$$
Both functions are <smooth function>[smooth] on their respective half-lines, proving the required assertion.
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