Evaluate the Integral

Problem

(∫_^)(((ln(x))2)/x*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of ln(x) is 1/x which is present in the integrand.

2. Substitute u=ln(x)

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

4. Rewrite the integral in terms of u by substituting the expressions from the previous steps.

(∫_^)(u2*d(u))

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

(u3)/3+C

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

Final Answer

(∫_^)(((ln(x))2)/x*d(x))=((ln(x))3)/3+C

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