Answer is: option4

\(\frac{1}{2}x \sin(2x) + \frac{1}{4} \cos(2x) + C\)

Solution:

To solve the integral \( \int x \cos(2x) \, dx \), we will use integration by parts.

Step 1: Differentiate and Integrate

We choose \( u = x \) and \( dv = \cos(2x) \, dx \).

Differentiate \( u \) and integrate \( dv \):

\[ u = x \quad \Rightarrow \quad du = dx \]

\[ dv = \cos(2x) \, dx \quad \Rightarrow \quad v = \frac{1}{2} \sin(2x) \]

Step 2: Apply the Integration by Parts Formula

The integration by parts formula is:

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

Substituting the values, we get:

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

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

Step 3: Solve the Remaining Integral

We compute \( \int \sin(2x) \, dx \):

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

Substituting this into the previous equation, we get:

\[ \frac{1}{2} x \sin(2x) - \frac{1}{2} \left( -\frac{1}{2} \cos(2x) \right) \]

\[ = \frac{1}{2} x \sin(2x) + \frac{1}{4} \cos(2x) \]

Step 4: Add the Constant of Integration

Finally, adding the constant of integration \( C \):

\[ \int x \cos(2x) \, dx = \frac{1}{2} x \sin(2x) + \frac{1}{4} \cos(2x) + C \]

Final Answer

\[ \frac{1}{2} x \sin(2x) + \frac{1}{4} \cos(2x) + C \]