Approximating function of the integral $\dfrac{\sin(x)^k}{x^k}$

Question

I needed a formula giving the value of the following integral; $$J(k)=\int_0^\pi dx\dfrac{\sin(x)^k}{x^k}$$ I didn't find any analytic expresion for it. I only found this interpolating function: $$J(k)=a\left(b^{\left(\dfrac{1}{k}\right)}\right)k^c-d$$ with: $a = 2.173,b = 0.8536,c = -0.4982,d = -0.002996$. valid until $k = 200$ with small error. Is t known any best fitting for $J(k)$? Thanks

Answer 1

After using the trigonomy formulas for $(\sin x)^n$ and applying partial integration, I've got with $n\in\mathbb{N}$:

$$J(n)=\int\limits_0^\pi\left(\frac{\sin x}{x}\right)^n dx = \frac{1}{(n-1)!}\sum\limits_{k=0}^{\lfloor (n-1)/2\rfloor}(-1)^k{\binom n k}\left(\frac{n}{2}-k\right)^{n-1}\text{Si}((n-2k)\pi)$$