Find the Integral ( natural log of x)/x

Problem

(∫_^)(ln(x)/x*d(x))

Solution

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

2. Substitute a new variable u for the logarithmic term.

u=ln(x)

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=1/x

d(u)=1/x*d(x)

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

(∫_^)(u*d(u))

5. Integrate with respect to u using the power rule for integration.

(u2)/2+C

6. Back-substitute the original expression for u to return the answer in terms of x

((ln(x))2)/2+C

Final Answer

(∫_^)(ln(x)/x*d(x))=((ln(x))2)/2+C

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