Evaluate the Integral

Problem

(∫_^)(ln(x)/x*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the derivative of ln(x) is 1/x which is present in the integrand.

2. Substitute u=ln(x)

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

4. Rewrite the integral in terms of u

(∫_^)(u4*d(u))

5. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u5)/5+C

6. Back-substitute u=ln(x) to return to the original variable x

Final Answer

(∫_^)(ln(x)/x*d(x))=ln(x)/5+C

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