Evaluate the Integral

Problem

(∫_25^3)((6*ln(y))/√(,y)*d(y))

Solution

1. Identify the integral as a definite integral and pull the constant 6 outside the integral.

(∫_25^3)((6*ln(y))/√(,y)*d(y))=6*(∫_25^3)(y(−1/2)*ln(y)*d(y))

2. Apply integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=ln(y) and d(v)=y(−1/2)*d(y)

d(u)=1/y*d(y)

v=2*y(1/2)

3. Substitute the parts into the integration by parts formula.

6*(∫_^)(y(−1/2)*ln(y)*d(y))=6*(2*y(1/2)*ln(y)−(∫_^)(2*y(1/2)⋅1/y*d(y)))

4. Simplify the remaining integral.

6*(2√(,y)*ln(y)−2*(∫_^)(y(−1/2)*d(y)))

6*(2√(,y)*ln(y)−2*(2*y(1/2)))

12√(,y)*ln(y)−24√(,y)

5. Evaluate the expression at the upper limit 3 and the lower limit 25

(12√(,3)*ln(3)−24√(,3))−(12√(,25)*ln(25)−24√(,25))

6. Simplify the numerical values, noting that √(,25)=5 and ln(25)=ln(5)=2*ln(5)

(12√(,3)*ln(3)−24√(,3))−(12*(5)*(2*ln(5))−24*(5))

12√(,3)*ln(3)−24√(,3)−120*ln(5)+120

Final Answer

(∫_25^3)((6*ln(y))/√(,y)*d(y))=12√(,3)*ln(3)−24√(,3)−120*ln(5)+120

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