Evaluate the Integral integral of xsin(x/2) with respect to x

Problem

(∫_^)(x*sin(x/2)*d(x))

Solution

1. Identify the method of integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the variables u=x and d(v)=sin(x/2)*d(x)

3. Differentiate u to find d(u)=d(x)

4. Integrate d(v) to find v=−2*cos(x/2)

5. Substitute these into the integration by parts formula.

(∫_^)(x*sin(x/2)*d(x))=−2*x*cos(x/2)−(∫_^)(−2*cos(x/2)*d(x))

6. Simplify the integral term by pulling out the constant.

(∫_^)(x*sin(x/2)*d(x))=−2*x*cos(x/2)+2*(∫_^)(cos(x/2)*d(x))

7. Evaluate the remaining integral.

(∫_^)(cos(x/2)*d(x))=2*sin(x/2)

8. Combine all terms and add the constant of integration C

−2*x*cos(x/2)+2*(2*sin(x/2))+C

Final Answer

(∫_^)(x*sin(x/2)*d(x))=−2*x*cos(x/2)+4*sin(x/2)+C

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