To solve the integral \(\int x \sin\left(\frac{x}{2}\right) \, dx\) using Integration by Parts (IBP), we follow these steps:

### Step 1: Identify parts for Integration by Parts (IBP)
Recall that Integration by Parts is given by:
\[
\int u \, dv = uv - \int v \, du
\]
For the integral \(\int x \sin\left(\frac{x}{2}\right) \, dx\), we choose:
- \(u = x\) (since its derivative simplifies)
- \(dv = \sin\left(\frac{x}{2}\right) dx\) (which we will integrate)

### Step 2: Compute \(du\) and \(v\)
Differentiate \(u\) and integrate \(dv\):
\[
du = dx
\]
\[
v = \int \sin\left(\frac{x}{2}\right) dx
\]
To integrate \(v\), let’s compute \(\int \sin\left(\frac{x}{2}\right) dx\):
- Use substitution: Let \(w = \frac{x}{2}\), hence \(dw = \frac{1}{2} dx\).
\[
\int \sin\left(\frac{x}{2}\right) dx = 2 \int \sin(w) dw = -2 \cos(w) + C = -2 \cos\left(\frac{x}{2}\right)
\]
So, \(v = -2 \cos\left(\frac{x}{2}\right)\).

### Step 3: Apply the Integration by Parts formula
Now, apply the formula:
\[
\int x \sin\left(\frac{x}{2}\right) \, dx = u \cdot v - \int v \, du
\]
Substitute the values of \(u\), \(v\), and \(du\):
\[
\int x \sin\left(\frac{x}{2}\right) \, dx = x \cdot \left(-2 \cos\left(\frac{x}{2}\right)\right) - \int \left(-2 \cos\left(\frac{x}{2}\right)\right) \, dx
\]
Simplify:
\[
= -2x \cos\left(\frac{x}{2}\right) + 2 \int \cos\left(\frac{x}{2}\right) \, dx
\]

### Step 4: Integrate the remaining integral
Integrate \(\int \cos\left(\frac{x}{2}\right) \, dx\):
- Using the same substitution \(w = \frac{x}{2}\), \(dw = \frac{1}{2} dx\):
\[
\int \cos\left(\frac{x}{2}\right) dx = 2 \int \cos(w) dw = 2 \sin(w) + C = 2 \sin\left(\frac{x}{2}\right)
\]
So the remaining integral is:
\[
2 \int \cos\left(\frac{x}{2}\right) dx = 4 \sin\left(\frac{x}{2}\right)
\]

### Step 5: Write the final answer
Combining all parts:
\[
\int x \sin\left(\frac{x}{2}\right) \, dx = -2x \cos\left(\frac{x}{2}\right) + 4 \sin\left(\frac{x}{2}\right) + C
\]
where \(C\) is the constant of integration.

Would you like any details on these steps, or do you have any questions?

Here are 5 related questions for further practice:
1. Integrate \(\int x \cos\left(\frac{x}{2}\right) \, dx\) using Integration by Parts.
2. Evaluate \(\int x e^{\frac{x}{2}} \, dx\) using Integration by Parts.
3. Solve the integral \(\int x^2 \sin\left(x\right) \, dx\) using Integration by Parts.
4. Integrate \(\int \sin\left(\frac{x}{3}\right) \, dx\) using substitution.
5. Find the integral \(\int x^2 e^{x} \, dx\) using Integration by Parts.

**Tip:** When applying Integration by Parts, always aim to simplify the integrand with your choice of \(u\) and \(dv\).

Learn how to solve the integral \(\int x \sin\left(\frac{x}{2}\right) \, dx\) using Integration by Parts (IBP). This detailed step-by-step solution covers key concepts such as integration techniques and trigonometric integrals. Suitable for advanced undergraduate students and anyone preparing for calculus examinations.

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