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 $dv$.