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We are tasked with evaluating the indefinite integral:

\[
\int x \sin(2x) \, dx
\]

The problem suggests using integration by parts. The general formula for integration by parts is:

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

Given:
- \( u = x \)
- \( dv = \sin(2x) \, dx \)

### Step-by-Step Solution:

1. **Differentiate \( u \)**:
   \[
   u = x \quad \Rightarrow \quad du = dx
   \]

2. **Integrate \( dv \)**:
   \[
   dv = \sin(2x) \, dx \quad \Rightarrow \quad v = \int \sin(2x) \, dx
   \]
   To integrate \( \sin(2x) \), we use a simple substitution or recall the known integral:
   \[
   \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x)
   \]

3. **Apply the integration by parts formula**:
   \[
   \int x \sin(2x) \, dx = uv - \int v \, du
   \]
   Substituting:
   \[
   \int x \sin(2x) \, dx = x \left( -\frac{1}{2} \cos(2x) \right) - \int \left( -\frac{1}{2} \cos(2x) \right) \, dx
   \]
   Simplifying:
   \[
   = -\frac{1}{2} x \cos(2x) + \frac{1}{2} \int \cos(2x) \, dx
   \]

4. **Solve the remaining integral**:
   \[
   \int \cos(2x) \, dx = \frac{1}{2} \sin(2x)
   \]
   So the full expression becomes:
   \[
   \int x \sin(2x) \, dx = -\frac{1}{2} x \cos(2x) + \frac{1}{4} \sin(2x) + C
   \]

### Now, let's analyze the given options:

Looking at the provided choices, the correct form should match this structure before integrating the second term. The correct answer is:

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

This matches the fourth option:

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

Would you like further details or explanations on any step? Here are some additional related questions for practice:

1. What is the integral of \( \sin(3x) \)?
2. How would you apply integration by parts to \( \int x e^{2x} \, dx \)?
3. What is the result of \( \int x^2 \sin(2x) \, dx \)?
4. How does the integration by parts formula change when integrating functions with powers of \( x \)?
5. How do you integrate \( \int x^n e^{kx} \, dx \)?

**Tip:** When applying integration by parts, always ensure you choose \( u \) and \( dv \) such that \( du \) simplifies and \( v \) is easy to integrate.

Consider the indefinite integral ∫ x sin(2x) dx. If we integrate by parts with u = x and dv = sin(2x) dx, which of the following is the integral above equal to?

Solve the indefinite integral ∫ x sin(2x) dx using integration by parts. This step-by-step solution covers the formula ∫ u dv = uv - ∫ v du and includes the integration of trigonometric functions such as sin(2x).

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