Evaluate the Integral

Problem

(∫_^)(ln(x)/x*d(x))

Solution

1. Identify the structure of the integral and notice that the derivative of ln(x) is 1/x which suggests using usubstitution.

2. Substitute u=ln(x) which implies that the derivative is d(u)/d(x)=1/x

3. Rewrite the differential d(x) in terms of d(u) as d(u)=1/x*d(x)

4. Substitute these values into the original integral to transform it into a power rule problem.

(∫_^)(u7*d(u))

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

(u8)/8+C

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

ln(x)/8+C

Final Answer

(∫_^)(ln(x)/x*d(x))=ln(x)/8+C

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