6. The indefinite integral \( \int 4 x \sin (2 x) \mathrm{d} x \) equals

Calculate the integral \( \int 4x \sin(2x) dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int 4x\sin\left(2x\right) dx\) - step1: Use properties of integrals: \(4\times \int x\sin\left(2x\right) dx\) - step2: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=\sin\left(2x\right)dx\end{align}\) - step3: Calculate the derivative: \(\begin{align}&du=dx\\&dv=\sin\left(2x\right)dx\end{align}\) - step4: Evaluate the integral: \(\begin{align}&du=dx\\&v=-\frac{1}{2}\cos\left(2x\right)\end{align}\) - step5: Substitute the values into formula: \(4\left(x\left(-\frac{1}{2}\cos\left(2x\right)\right)-\int 1\times \left(-\frac{1}{2}\cos\left(2x\right)\right) dx\right)\) - step6: Calculate: \(4\left(-\frac{1}{2}x\cos\left(2x\right)-\int -\frac{1}{2}\cos\left(2x\right) dx\right)\) - step7: Calculate: \(-2x\cos\left(2x\right)-4\times \int -\frac{1}{2}\cos\left(2x\right) dx\) - step8: Evaluate the integral: \(-2x\cos\left(2x\right)+\sin\left(2x\right)\) - step9: Add the constant of integral C: \(-2x\cos\left(2x\right)+\sin\left(2x\right) + C, C \in \mathbb{R}\) The indefinite integral \( \int 4x \sin(2x) dx \) equals \( -2x\cos(2x) + \sin(2x) + C \), where \( C \) is an arbitrary constant.