Answer is: option2

\(\frac{1}{3} x^2 \sin(3x)\)

Solution:

We are given the equation:

\[ \int x^2 \cos(3x) \, dx = f(x) - \frac{2}{3} \int x \sin(3x) \, dx \]

We need to determine \( f(x) \).

Step 1: Solve \( \int x^2 \cos(3x) \, dx \) using Integration by Parts

We will solve \( \int x^2 \cos(3x) \, dx \) using integration by parts. Let us choose:

\[ u = x^2 \quad \Rightarrow \quad du = 2x \, dx \]

\[ dv = \cos(3x) \, dx \quad \Rightarrow \quad v = \frac{1}{3} \sin(3x) \]

Step 2: Apply Integration by Parts

Now, apply the integration by parts formula:

\[ \int u \, dv = u v - \int v \, du \]

Substitute the values of \( u \), \( du \), \( v \), and \( dv \):

\[ \int x^2 \cos(3x) \, dx = x^2 \cdot \frac{1}{3} \sin(3x) - \int 2x \cdot \frac{1}{3} \sin(3x) \, dx \]

Simplify the expression:

\[ \int x^2 \cos(3x) \, dx = \frac{1}{3} x^2 \sin(3x) - \frac{2}{3} \int x \sin(3x) \, dx \]

Step 3: Compare with the Given Equation

Now, comparing this result with the given equation:

\[ \int x^2 \cos(3x) \, dx = f(x) - \frac{2}{3} \int x \sin(3x) \, dx \]

We can see that:

\[ f(x) = \frac{1}{3} x^2 \sin(3x) \]

Final Answer

The function \( f(x) \) is:

\[ f(x) = \frac{1}{3} x^2 \sin(3x) \]