Evaluate the Integral

Problem

(∫_1^3)(((ln(x))2)/(x3)*d(x))

Solution

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

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

3. Differentiate u to find d(u)=2*ln(x)⋅1/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 again for the new integral (∫_^)(ln(x)/(x3)*d(x)) by letting u=ln(x) and d(v)=x(−3)*d(x)

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

8. Substitute these into the formula:

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

9. Evaluate the remaining integral:

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)+(∫_^)(1/2*x(−3)*d(x))

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

10. 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)

11. Evaluate the definite integral from 1 to 3

(∫_1^3)(((ln(x))2)/(x3)*d(x))=[−(2*(ln(x))2+2*ln(x)+1)/(4*x2)]31

12. Substitute the upper and lower limits:

(−(2*(ln(3))2+2*ln(3)+1)/(4*(3)2))−(−(2*(ln(1))2+2*ln(1)+1)/(4*(1)2))

13. Simplify using ln(1)=0

−(2*(ln(3))2+2*ln(3)+1)/36+1/4

14. Combine into a single fraction:

(9−(2*(ln(3))2+2*ln(3)+1))/36

(8−2*ln(3)−2*(ln(3))2)/36

(4−ln(3)−(ln(3))2)/18

Final Answer

(∫_1^3)(((ln(x))2)/(x3)*d(x))=(4−ln(3)−(ln(3))2)/18

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