Integral with Fourier transform

Question

Let $$f(y)=\int_{-\infty}^{\infty}\frac{\exp(-2\pi ixy)}{1+x^{2q}}dx $$ How using the fact that $f(y)$ is Fourier transform of $\frac{1}{1+x^{2q}} $ to show that $$\int_{-\infty}^{\infty}f(x)x^{u}dx=0,\ u=1,...2q-1 $$

Answer 1

Hint: use these properties of Fourier transform (106 here), and inverse transform at $x=0$.

$$g(t) \longleftrightarrow G(\omega) $$ $$ \hspace{58px} ? \; \longleftrightarrow \int_{-\infty}^{\infty} G(\omega) d\omega $$ $$\hspace{28px} ? \; \longleftrightarrow \omega^n G(\omega) $$