Evaluate the Integral

Problem

(∫_1^7)(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 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 integration by parts formula:

(∫_^)(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 7

[(w3)/3*ln(w)−(w3)/9]71

8. Substitute the upper limit w=7

7/3*ln(7)−7/9=343/3*ln(7)−343/9

9. Substitute the lower limit w=1

1/3*ln(1)−1/9=0−1/9=−1/9

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

343/3*ln(7)−343/9−(−1/9)

11. Simplify the final expression:

343/3*ln(7)−342/9

12. Reduce the fraction 342/9=38

343/3*ln(7)−38

Final Answer

(∫_1^7)(w2*ln(w)*d(w))=343/3*ln(7)−38

Want more problems? Check here!