Evaluate the Integral

Problem

(∫_^)(ln(2*x)/x*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of ln(2*x) is proportional to 1/x

2. Substitute u=ln(2*x)

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

d(u)/d(x)=1/(2*x)⋅2

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 using the power rule for integration.

(u2)/2+C

6. Back-substitute the original expression for u

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

Final Answer

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

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