 Solution

There is an important convention issue in the printed statement. The ordinary Cauchy principal value exists only for $n=1$. Closing the Fourier contour in the lower half-plane, with a small indentation above the real-axis pole, givesEquivalently, the odd cosine contribution cancels and the Dirichlet integral ${\int }_0^{\infty }\sin x/xdx=\pi /2$ supplies the imaginary part.

For $n\ge 2$, the paired integrand near zero is $2\cos x/x^n$ if $n$ is even and $-2i\sin x/x^n$ if $n$ is odd. Its leading even power is nonintegrable: $x^{-n}$ in the first case and $x^{1-n}$ in the second. Thus the ordinary principal value diverges for every $n>1$.

If the intended convention is the Hadamard finite-part integral, define it consistently by distributional differentiation,Integration by parts in this distributional pairing, or regularized Fourier transformation, then givesThe distinction is essential: subtracting the divergent local Taylor terms supplies a finite part, but symmetric exclusion alone does not.