Evaluate
\left\{\begin{matrix}\frac{-k^{3}\sin(x\left(k+2\right))+k^{3}\sin(x\left(k-2\right))+2k^{3}\sin(2x)\cos(kx)+2k^{2}\sin(x\left(k+2\right))+2k^{2}\sin(x\left(k-2\right))-8k\sin(2x)\cos(kx)+4k^{2}\sin(kx)-16\sin(kx)}{8k\left(k^{2}-4\right)}+С,&k\neq 0\text{ and }|k|\neq 2\\\frac{\sin(2x)}{4}+\frac{\sin(4x)}{16}+\frac{x}{4}+С,&|k|=2\\\frac{\sin(2x)}{4}+\frac{x}{2}+С,&k=0\end{matrix}\right.
Differentiate w.r.t. x
\left\{\begin{matrix}\frac{-k\cos(x\left(k+2\right))+k\cos(x\left(k-2\right))-2k\sin(2x)\sin(kx)+4\cos(2x)\cos(kx)+4\cos(kx)}{8},&|k|\neq 2\\\frac{\cos(4x)+2\cos(2x)+1}{4},&|k|=2\end{matrix}\right.
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