Evaluate the Integral

Problem

(∫_^)(p5*ln(p)*d(p))

Solution

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

2. Choose u=ln(p) and d(v)=p5*d(p) to simplify the expression through differentiation.

3. Differentiate u to find d(u)=1/p*d(p)

4. Integrate d(v) to find v=(p6)/6

5. Substitute these values into the integration by parts formula.

(∫_^)(p5*ln(p)*d(p))=ln(p)⋅(p6)/6−(∫_^)((p6)/6⋅1/p*d(p))

6. Simplify the integral on the right side.

(∫_^)(p5*ln(p)*d(p))=(p6*ln(p))/6−(∫_^)((p5)/6*d(p))

7. Evaluate the remaining integral.

(∫_^)((p5)/6*d(p))=(p6)/36

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

Final Answer

(∫_^)(p5*ln(p)*d(p))=(p6*ln(p))/6−(p6)/36+C

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