Evaluate the Integral

Problem

(∫_4^9)(ln(y)/√(,y)*d(y))

Solution

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

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

3. Differentiate u to find d(u)=1/y*d(y) and integrate d(v) to find 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 integral on the right side.

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

6. Evaluate the remaining integral.

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

7. Apply the limits of integration from 4 to 9 using the Fundamental Theorem of Calculus.

(∫_4^9)(ln(y)/√(,y)*d(y))=[2√(,y)*ln(y)−4√(,y)]94

8. Substitute the upper limit y=9

2√(,9)*ln(9)−4√(,9)=6*ln(9)−12

9. Substitute the lower limit y=4

2√(,4)*ln(4)−4√(,4)=4*ln(4)−8

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

(6*ln(9)−12)−(4*ln(4)−8)=6*ln(9)−4*ln(4)−4

11. Simplify the logarithms using the property ln(ab)=b*ln(a)

6*ln(3)−4*ln(2)−4=12*ln(3)−8*ln(2)−4

Final Answer

(∫_4^9)(ln(y)/√(,y)*d(y))=12*ln(3)−8*ln(2)−4

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