Evaluate the Integral

Problem

(∫_1^5)(((ln(x))2)/(x3)*d(x))

Solution

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

2. Set the variables for the first application of integration by parts where u=(ln(x))2 and d(v)=x(−3)*d(x)

3. Differentiate u to find d(u)=(2*ln(x))/x*d(x) and integrate d(v) to find v=−1/(2*x2)

4. Apply the integration by parts formula:

(∫_^)(((ln(x))2)/(x3)*d(x))=−((ln(x))2)/(2*x2)−(∫_^)(−1/(2*x2)⋅(2*ln(x))/x*d(x))

5. Simplify the resulting integral:

(∫_^)(((ln(x))2)/(x3)*d(x))=−((ln(x))2)/(2*x2)+(∫_^)(ln(x)/(x3)*d(x))

6. Apply integration by parts a second time for the new integral, setting u=ln(x) and d(v)=x(−3)*d(x) which gives d(u)=1/x*d(x) and v=−1/(2*x2)

7. Substitute these values into the second integration:

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)−(∫_^)(−1/(2*x2)⋅1/x*d(x))

8. Evaluate the final integral:

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)+(∫_^)(1/(2*x3)*d(x))

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)−1/(4*x2)

9. Combine all parts to find the general antiderivative:

(∫_^)(((ln(x))2)/(x3)*d(x))=−((ln(x))2)/(2*x2)−ln(x)/(2*x2)−1/(4*x2)

10. Evaluate the definite integral from 1 to 5 by substituting the bounds:

[−(2*(ln(x))2+2*ln(x)+1)/(4*x2)]51

11. Calculate the value at the upper bound x=5

−(2*(ln(5))2+2*ln(5)+1)/100

12. Calculate the value at the lower bound x=1

−(2*(ln(1))2+2*ln(1)+1)/4=−1/4

13. Subtract the lower bound value from the upper bound value:

−(2*(ln(5))2+2*ln(5)+1)/100−(−1/4)

(25−2*(ln(5))2−2*ln(5)−1)/100

(24−2*ln(5)−2*(ln(5))2)/100

14. Simplify the final fraction:

(12−ln(5)−(ln(5))2)/50

Final Answer

(∫_1^5)(((ln(x))2)/(x3)*d(x))=(12−ln(5)−(ln(5))2)/50

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