Evaluate the Integral

Problem

(∫_64^8)(ln(y)/√(,y)*d(y))

Solution

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

2. Assign variables for integration by parts. Let u=ln(y) and d(v)=y(−1/2)*d(y)

3. Differentiate u and integrate d(v) This gives d(u)=1/y*d(y) and v=2√(,y)

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

(∫_^)(ln(y)/√(,y)*d(y))=2√(,y)*ln(y)−(∫_^)(2√(,y)⋅1/y*d(y))

5. Simplify the remaining integral.

(∫_^)(ln(y)/√(,y)*d(y))=2√(,y)*ln(y)−2*(∫_^)(y(−1/2)*d(y))

6. Evaluate the indefinite integral.

(∫_^)(ln(y)/√(,y)*d(y))=2√(,y)*ln(y)−4√(,y)+C

7. Apply the Fundamental Theorem of Calculus using the limits 64 and 8

(∫_64^8)(ln(y)/√(,y)*d(y))=[2√(,y)*ln(y)−4√(,y)]864

8. Substitute the upper limit y=8

2√(,8)*ln(8)−4√(,8)=4√(,2)*ln(2)−8√(,2)=12√(,2)*ln(2)−8√(,2)

9. Substitute the lower limit y=64

2√(,64)*ln(64)−4√(,64)=16*ln(2)−32=96*ln(2)−32

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

(12√(,2)*ln(2)−8√(,2))−(96*ln(2)−32)

Final Answer

(∫_64^8)(ln(y)/√(,y)*d(y))=12√(,2)*ln(2)−8√(,2)−96*ln(2)+32

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