Evaluate the Integral

Problem

(∫_1^5)(w2*ln(w)*d(w))

Solution

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

2. Assign the variables for integration by parts by letting u=ln(w) and d(v)=w2*d(w)

3. Differentiate u to find d(u)=1/w*d(w) and integrate d(v) to find v=(w3)/3

4. Apply the formula for integration by parts to the indefinite integral.

(∫_^)(w2*ln(w)*d(w))=(w3)/3*ln(w)−(∫_^)((w3)/3⋅1/w*d(w))

5. Simplify the integral on the right.

(∫_^)(w2*ln(w)*d(w))=(w3)/3*ln(w)−(∫_^)((w2)/3*d(w))

6. Evaluate the remaining integral.

(∫_^)(w2*ln(w)*d(w))=(w3)/3*ln(w)−(w3)/9

7. Apply the limits of integration from 1 to 5

(∫_1^5)(w2*ln(w)*d(w))=[(w3)/3*ln(w)−(w3)/9]51

8. Substitute the upper limit w=5

5/3*ln(5)−5/9=125/3*ln(5)−125/9

9. Substitute the lower limit w=1

1/3*ln(1)−1/9=0−1/9

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

125/3*ln(5)−125/9−(−1/9)=125/3*ln(5)−124/9

Final Answer

(∫_1^5)(w2*ln(w)*d(w))=125/3*ln(5)−124/9

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