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:
$\frac{\sin{kx}}{k}$
$\frac{\cos{kx}}{k^2}+\frac{x \sin{kx}}{k}$
$\frac{2 x \cos{kx}}{k^2}+\frac{\left(-2+k^2 x^2\right) \sin{kx}}{k^3}$
$\frac{3 \left(-2+k^2 x^2\right) \cos{kx}}{k^4}+\frac{x \left(-6+k^2 x^2\right) \sin{kx}}{k^3}$
$\frac{4 x \left(-6+k^2 x^2\right) \cos{kx}}{k^4}+\frac{\left(24-12 k^2 x^2+k^4 x^4\right) \sin{kx}}{k^5}$
$\frac{5 \left(24-12 k^2 x^2+k^4 x^4\right) \cos{kx}}{k^6}+\frac{x \left(120-20 k^2 x^2+k^4 x^4\right) \sin{kx}}{k^5}$
$\frac{6 x \left(120-20 k^2 x^2+k^4 x^4\right) \cos{kx}}{k^6}+\frac{\left(-720+360 k^2 x^2-30 k^4 x^4+k^6 x^6\right) \sin{kx}}{k^7}$