Evaluate the Integral

Problem

(∫_^)(ln(x)/x*d(x))

Solution

1. Identify the structure of the integrand, noting that the derivative of ln(x) is 1/x which suggests using usubstitution.

2. Substitute u=ln(x) which implies that d(u)=1/x*d(x)

3. Rewrite the integral in terms of u

(∫_^)(u3*d(u))

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

(u4)/4+C

5. Back-substitute the original expression for u to get the final result in terms of x

ln(x)/4+C

Final Answer

(∫_^)(ln(x)/x*d(x))=ln(x)/4+C

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