a) \( \int x \operatorname{sen}\left(\frac{x}{3}\right) d x= \)

Calculate or simplify the expression \( \int x \sin\left(\frac{x}{3}\right) dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x\sin\left(\frac{x}{3}\right) dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=\sin\left(\frac{x}{3}\right)dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=dx\\&dv=\sin\left(\frac{x}{3}\right)dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=dx\\&v=-3\cos\left(\frac{1}{3}x\right)\end{align}\) - step4: Substitute the values into formula: \(x\left(-3\cos\left(\frac{1}{3}x\right)\right)-\int 1\times \left(-3\cos\left(\frac{1}{3}x\right)\right) dx\) - step5: Calculate: \(-3x\cos\left(\frac{1}{3}x\right)-\int -3\cos\left(\frac{1}{3}x\right) dx\) - step6: Evaluate the integral: \(-3x\cos\left(\frac{1}{3}x\right)+9\sin\left(\frac{1}{3}x\right)\) - step7: Add the constant of integral C: \(-3x\cos\left(\frac{1}{3}x\right)+9\sin\left(\frac{1}{3}x\right) + C, C \in \mathbb{R}\) La integral de \(x \sin\left(\frac{x}{3}\right)\) con respecto a \(x\) es igual a \(-3x\cos\left(\frac{x}{3}\right) + 9\sin\left(\frac{x}{3}\right) + C\), donde \(C\) es una constante real.