Polynomials in Fourier trigonometric series I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding: 
$\frac{\sin{kx}}{k}$, 
$\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$, 
$\frac{2 x \cos{kx}}{k^2}+\frac{\left(-2+k^2 x^2\right)sin{kx}}{k^3}$, 
$\frac{3 \left(-2+k^2 x^2\right) \cos{kx}}{k^4}+\frac{x \left(-6+k^2 x^2\right) \sin(kx)}{k^3}$
Is there a name for the sequence of polynomials: $x$, $2x$, $k^2x^2-2$, $3(k^2x^2-2)$, $x(k^2x^2-6)$ ... ?
Here is more:
$\frac{\sin{kx}}{k}$
$\frac{\cos{kx}}{k^2}+\frac{x \sin{kx}}{k}$
$\frac{2 x \cos{kx}}{k^2}+\frac{\left(-2+k^2 x^2\right) \sin{kx}}{k^3}$
$\frac{3 \left(-2+k^2 x^2\right) \cos{kx}}{k^4}+\frac{x \left(-6+k^2 x^2\right) \sin{kx}}{k^3}$
$\frac{4 x \left(-6+k^2 x^2\right) \cos{kx}}{k^4}+\frac{\left(24-12 k^2 x^2+k^4 x^4\right) \sin{kx}}{k^5}$
$\frac{5 \left(24-12 k^2 x^2+k^4 x^4\right) \cos{kx}}{k^6}+\frac{x \left(120-20 k^2 x^2+k^4 x^4\right) \sin{kx}}{k^5}$
$\frac{6 x \left(120-20 k^2 x^2+k^4 x^4\right) \cos{kx}}{k^6}+\frac{\left(-720+360 k^2 x^2-30 k^4 x^4+k^6 x^6\right) \sin{kx}}{k^7}$

You could also consider the exponential generating function
$$ \sum_{n=0}^\infty \dfrac{t^n}{n!} \int_0^x s^n e^{iks}\ ds = \int_0^x e^{(t+ik)s}\ ds = \dfrac{e^{(t+ik)x} - 1}{t+ik}$$
This is the product of $e^{(t+ik)x}-1 = -1 + \sum_{n=0}^\infty \dfrac{t^n x^n}{n!} e^{ikx}$ and $\dfrac{1}{t+ik} = \sum_{n=0}^\infty (-1)^n \dfrac{t^n}{(ik)^{n+1}}$, so 
$$\int_0^x s^n e^{iks}\ ds = n! \left(\dfrac{1}{(-ik)^{n+1}} + \sum_{j=0}^n e^{ikx} \dfrac{(-1)^{n-j}}{j! (ik)^{n-j+1}} x^j\right)$$
