Evaluate the Integral

Problem

(∫_^)(ln(2*x)*d(x))

Solution

1. Identify the method of integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign the variables for integration by parts by letting u=ln(2*x) and d(v)=d(x)

3. Differentiate u to find d(u)=1/(2*x)⋅2*d(x)=1/x*d(x) and integrate d(v) to find v=x

4. Substitute these values into the integration by parts formula.

(∫_^)(ln(2*x)*d(x))=x*ln(2*x)−(∫_^)(x⋅1/x*d(x))

5. Simplify the integrand in the second term.

(∫_^)(ln(2*x)*d(x))=x*ln(2*x)−(∫_^)(1*d(x))

6. Integrate the remaining term and add the constant of integration C

(∫_^)(ln(2*x)*d(x))=x*ln(2*x)−x+C

Final Answer

(∫_^)(ln(2*x)*d(x))=x*ln(2*x)−x+C

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