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

To solve the integral \( \int x \sin\left(\frac{x}{3}\right) dx \), we can use integration by parts. Let \( u = x \) and \( dv = \sin\left(\frac{x}{3}\right) dx \). Then, \( du = dx \) and \( v = -3 \cos\left(\frac{x}{3}\right) \) (since we integrate \( \sin\left(\frac{x}{3}\right) \) to get \(-3\cos\left(\frac{x}{3}\right)\)). Applying integration by parts: \[ \int x \sin\left(\frac{x}{3}\right) dx = uv - \int v \, du \] This gives us: \[ = -3x \cos\left(\frac{x}{3}\right) - \int (-3 \cos\left(\frac{x}{3}\right)) dx \] Now we integrate \( -3 \cos\left(\frac{x}{3}\right) \): \[ = -3x \cos\left(\frac{x}{3}\right) + 9 \sin\left(\frac{x}{3}\right) + C \] Thus, the final result for the integral is: \[ \int x \sin\left(\frac{x}{3}\right) dx = -3x \cos\left(\frac{x}{3}\right) + 9 \sin\left(\frac{x}{3}\right) + C \]