Evaluate the Integral

Problem

(∫_^)(ln(x)/(2*x)*d(x))

Solution

1. Rewrite the integral by pulling out the constant factor to make the structure clearer.

1/2*(∫_^)(ln(x)/x*d(x))

2. Identify a substitution by letting u=ln(x) since its derivative 1/x is present in the integrand.

u=ln(x)

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

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

4. Substitute u and d(u) into the integral.

1/2*(∫_^)(u*d(u))

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

1/2⋅(u2)/2+C

6. Simplify the expression.

(u2)/4+C

7. Back-substitute u=ln(x) to return to the original variable.

((ln(x))2)/4+C

Final Answer

(∫_^)(ln(x)/(2*x)*d(x))=((ln(x))2)/4+C

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