Evaluate the Integral

Problem

(∫_36^4)((9*ln(y))/√(,y)*d(y))

Solution

1. Simplify the integral by pulling the constant factor outside the integral sign.

(∫_36^4)((9*ln(y))/√(,y)*d(y))=9*(∫_36^4)(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.

9*(∫_^)(y(−1/2)*ln(y)*d(y))=9*(2*y(1/2)*ln(y)−(∫_^)(2*y(1/2)⋅1/y*d(y)))

4. Simplify the remaining integral inside the expression.

9*(2√(,y)*ln(y)−2*(∫_^)(y(−1/2)*d(y)))

9*(2√(,y)*ln(y)−2*(2*y(1/2)))

18√(,y)*ln(y)−36√(,y)

5. Evaluate the definite integral by applying the limits of integration from 36 to 4

[18√(,y)*ln(y)−36√(,y)]436

6. Calculate the value at the upper limit y=4

18√(,4)*ln(4)−36√(,4)=36*ln(4)−72

7. Calculate the value at the lower limit y=36

18√(,36)*ln(36)−36√(,36)=108*ln(36)−216

8. Subtract the lower limit value from the upper limit value.

(36*ln(4)−72)−(108*ln(36)−216)

36*ln(2)−72−108*ln(6)+216

72*ln(2)−216*ln(6)+144

9. Simplify the logarithmic terms using ln(6)=ln(2)+ln(3)

72*ln(2)−216*(ln(2)+ln(3))+144

72*ln(2)−216*ln(2)−216*ln(3)+144

−144*ln(2)−216*ln(3)+144

Final Answer

(∫_36^4)((9*ln(y))/√(,y)*d(y))=144−144*ln(2)−216*ln(3)

Want more problems? Check here!