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 the method of substitution.

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

3. Rewrite the integral in terms of u by replacing ln(x) with u and 1/x*d(x) with d(u)

(∫_^)(u6*d(u))

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

(u7)/7+C

5. Back-substitute the original expression for u to return to the variable x

ln(x)/7+C

Final Answer

(∫_^)(ln(x)/x*d(x))=ln(x)/7+C

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