Use integration by parts to evaluate the integral. \[ \int 4 x \ln (2 x) d x \]

Calculate the integral \( \int 4x \ln(2x) dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int 4x\ln{\left(2x\right)} dx\) - step1: Use properties of integrals: \(4\times \int x\ln{\left(2x\right)} dx\) - step2: Prepare for integration by parts: \(\begin{align}&u=\ln{\left(2x\right)}\\&dv=xdx\end{align}\) - step3: Calculate the derivative: \(\begin{align}&du=\frac{1}{x}dx\\&dv=xdx\end{align}\) - step4: Evaluate the integral: \(\begin{align}&du=\frac{1}{x}dx\\&v=\frac{x^{2}}{2}\end{align}\) - step5: Substitute the values into formula: \(4\left(\ln{\left(2x\right)}\times \frac{x^{2}}{2}-\int \frac{1}{x}\times \frac{x^{2}}{2} dx\right)\) - step6: Calculate: \(4\left(\frac{x^{2}\ln{\left(2x\right)}}{2}-\int \frac{x}{2} dx\right)\) - step7: Calculate: \(4\times \frac{x^{2}\ln{\left(2x\right)}}{2}-4\times \int \frac{x}{2} dx\) - step8: Evaluate the integral: \(4\times \frac{x^{2}\ln{\left(2x\right)}}{2}-x^{2}\) - step9: Reduce the fraction: \(2x^{2}\ln{\left(2x\right)}-x^{2}\) - step10: Add the constant of integral C: \(2x^{2}\ln{\left(2x\right)}-x^{2} + C, C \in \mathbb{R}\) The integral of \(4x \ln(2x) dx\) using integration by parts is \(2x^{2}\ln{(2x)}-x^{2} + C\), where \(C\) is an arbitrary constant.