Evaluate the Integral

Problem

(∫_^)(ln(5*x)/x*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the derivative of ln(5*x) is proportional to 1/x

2. Substitute u=ln(5*x)

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=1/(5*x)⋅5

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

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

4. Rewrite the integral in terms of u

(∫_^)(u*d(u))

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

(u2)/2+C

6. Back-substitute u=ln(5*x) to return to the original variable x

((ln(5*x))2)/2+C

Final Answer

(∫_^)(ln(5*x)/x*d(x))=((ln(5*x))2)/2+C

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