Evaluate the Integral

Problem

(∫_^)(ln(x)/(x2)*d(x))

Solution

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

2. Assign variables for the parts, choosing u=ln(x) and d(v)=1/(x2)*d(x)

3. Differentiate u to find d(u)=1/x*d(x)

4. Integrate d(v) to find v=−1/x

5. Substitute these into the integration by parts formula.

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

6. Simplify the integral on the right side.

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

7. Evaluate the remaining integral.

(∫_^)(1/(x2)*d(x))=−1/x

8. Combine the terms and add the constant of integration C

(∫_^)(ln(x)/(x2)*d(x))=−ln(x)/x−1/x+C

Final Answer

(∫_^)(ln(x)/(x2)*d(x))=−(ln(x)+1)/x+C

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