Evaluate the Integral

Problem

(∫_0^π)(x*cos(x)*d(x))

Solution

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

2. Assign the variables for integration by parts by letting u=x and d(v)=cos(x)*d(x)

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

4. Apply the integration by parts formula to the definite integral.

(∫_0^π)(x*cos(x)*d(x))=[x*sin(x)]π0−(∫_0^π)(sin(x)*d(x))

5. Evaluate the first term [x*sin(x)]π0 at the boundaries.

[x*sin(x)]π0=π*sin(π)−0*sin(0)

[x*sin(x)]π0=π*(0)−0*(0)=0

6. Integrate the remaining term (∫_^)(sin(x)*d(x)) which results in −cos(x)

−(∫_0^π)(sin(x)*d(x))=−[−cos(x)]π0

−(∫_0^π)(sin(x)*d(x))=[cos(x)]π0

7. Substitute the boundaries into the evaluated integral.

[cos(x)]π0=cos(π)−cos(0)

[cos(x)]π0=−1−1=−2

Final Answer

(∫_0^π)(x*cos(x)*d(x))=−2

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