= Solution
Integrate $e^{iz}/z$ around an upper semicircle, indenting the origin by a small upper semicircle traversed clockwise. The integrand is holomorphic inside the indented contour. The large arc vanishes by <Jordan's lemma>, and the small arc contributes $-i\pi$, so
$$
\operatorname{PV}\int_{-\infty}^{\infty}\frac{e^{ix}}x\,dx=i\pi.
$$
Taking imaginary and real parts yields
$$
\boxed{\int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx=\pi,\qquad
\operatorname{PV}\int_{-\infty}^{\infty}\frac{\cos x}{x}\,dx=0.}
$$
The first integrand has a removable singularity at zero and its tails converge conditionally by <Dirichlet test for improper integrals>. The second <integral> needs a <principal value> at zero; its zero also follows from oddness with symmetric exclusion. One must not assign an ordinary convergent <integral> to its singularity.
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