Evaluate the Integral

Problem

(∫_0^2*π)(x*sin(x)*d(x))

Solution

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

2. Assign variables for the parts: let u=x and d(v)=sin(x)*d(x)

3. Differentiate u to find d(u)=d(x) and integrate d(v) to find v=−cos(x)

4. Apply the formula for integration by parts:

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

5. Simplify the integral of the second term:

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

6. Evaluate the remaining integral:

(∫_^)(x*sin(x)*d(x))=−x*cos(x)+sin(x)

7. Apply the limits of integration from 0 to 2*π

[−x*cos(x)+sin(x)](2*π)0

8. Substitute the upper limit 2*π

−2*π*cos(2*π)+sin(2*π)=−2*π*(1)+0=−2*π

9. Substitute the lower limit 0

−0*cos(0)+sin(0)=0+0=0

10. Subtract the lower limit result from the upper limit result:

−2*π−0=−2*π

Final Answer

(∫_0^2*π)(x*sin(x)*d(x))=−2*π

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