I want to compute the following integral for an arbitrary $k \in \mathbb{N}$.
$$ \int_{-\infty}^{+\infty} \frac{\sin^k(x)}{x^k} dx$$
For $k = 2$ I computed:
$$ \int_{-\infty}^{+\infty} \frac{\sin^2(x)}{x^2} \, dx = $$
$$ \lim_{n\rightarrow\infty} \int_{-\frac{n\pi}{2}}^{+\frac{n\pi}{2}} \frac{\sin^2(x)}{x^2} \, dx = $$
(Substitute $x = \frac{nt}{2}$)
$$\lim_{n\rightarrow\infty} \int_{-\pi}^{\pi} \frac{\sin^2(\frac{nt}{2})}{(\frac{nt}{2})^2} \frac{n}{2} \, dt = $$
$$\lim_{n\rightarrow\infty} \int_{-\pi}^\pi \frac{2\sin^2(\frac{nt}{2})}{nt^2} \, dt = $$
$$\lim_{n\rightarrow\infty} \frac{1}{2\pi} \int_{-\pi}^\pi \frac{\pi\sin^2(\frac{t}{2})}{(\frac{t}{2})^2} \frac{1}{n} \left( \frac{\sin(\frac{nt}{2})}{\sin(\frac{t}{2})}\right)^2 \, dt = $$
$$\frac{f( 0^{+})- f(0^{-})}{2} = \pi$$
(where $\lim_{t\rightarrow0}\frac{\pi\sin^{2}(\frac{t}{2})}{(\frac{t}{2})^{2}} = \pi$ and $\frac{1}{n} (\frac{\sin(\frac{nt}{2})}{\sin(\frac{t}{2})})^{2}$ represents the Fourier-kernel)
Now for an arbitrary $k \in \mathbb{N}_{\geq3}$ I have tried to use the recursion formula for the sine, but I did not get very far.
I would appreciate any help or ideas!
