\( \int x\cos{(2x)}  dx \)=

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Use integration by parts:

$$ u= x \hskip{2em} dv= \cos{(2x)}  dx $$ $$ du=dx \hskip{2em} v= \frac{1}{2}\sin{(2x)} $$ $$ \int u   dv = uv - \int v   du $$ $$ \int x\cos{(2x)}  dx = x\left(\frac{1}{2}\sin{(2x)}\right) - \int \frac{1}{2}\sin{(2x)}   dx $$ $$ = \boxed{\frac{1}{2}x\sin{(2x)}+\frac{1}{4}\cos{(2x)}+C } $$