Evaluate the Integral

Problem

(∫_1^7)(ln(x)/(x3)*d(x))

Solution

1. Identify the integral as a definite integral requiring integration by parts, where u=ln(x) and d(v)=x(−3)*d(x)

2. Calculate the differentials for the first application of integration by parts: d(u)=(2*ln(x))/x*d(x) and v=−1/(2*x2)

3. Apply the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

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

4. Simplify the resulting integral:

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

5. Apply integration by parts again for the new integral, where u=ln(x) and d(v)=x(−3)*d(x) resulting in d(u)=1/x*d(x) and v=−1/(2*x2)

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

6. Evaluate the final integral term:

(∫_^)(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)

7. Combine all terms to find the general antiderivative:

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

8. Evaluate the definite integral from 1 to 7 by substituting the bounds:

[−(2*ln(x)+2*ln(x)+1)/(4*x2)]71

9. Substitute the upper bound x=7 and the lower bound x=1

(−(2*ln(7)+2*ln(7)+1)/(4*(7)2))−(−(2*ln(1)+2*ln(1)+1)/(4*(1)2))

10. Simplify using ln(1)=0

−(2*ln(7)+2*ln(7)+1)/196+1/4

11. Find a common denominator to combine the terms:

49/196−(2*ln(7)+2*ln(7)+1)/196

(48−2*ln(7)−2*ln(7))/196

12. Reduce the fraction by dividing by 2

(24−ln(7)−ln(7))/98

Final Answer

(∫_1^7)(ln(x)/(x3)*d(x))=(24−ln(7)−ln(7))/98

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