Integral of \( \displaystyle 2 x \sin{\left(2 x \right)} \)

Problem 4.883 · medium

1. \[ \int 2 x \sin{\left(2 x \right)}\, dx \]
   integralStart with the integral of the function.✓ Proved
2. \[ = \operatorname{parts}{\left(2 x \sin{\left(2 x \right)},x,1,\sin{\left(2 x \right)},- \frac{\cos{\left(2 x \right)}}{2} \right)} \]
   partsApply integration by parts where u = 2*x and dv = sin(2*x)dx.Reviewed
3. \[ = - x \cos{\left(2 x \right)} - \int \left(- \cos{\left(2 x \right)}\right)\, dx \]
   partsApply the integration by parts formula: integral(u dv) = u v - integral(v du).Reviewed
4. \[ = - x \cos{\left(2 x \right)} + \int \cos{\left(2 x \right)}\, dx \]
   simplifySimplify the expression inside and outside the integral.✓ Proved
5. \[ = - x \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{2} \]
   antiderivativeEvaluate the remaining integral.✓ Proved