Evaluate the Integral

Problem

(∫_^)(x*ln(2*x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(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)=x*d(x)

3. Differentiate u to find d(u)=1/x*d(x) and integrate d(v) to find v=(x2)/2

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

(∫_^)(x*ln(2*x)*d(x))=(x2)/2*ln(2*x)−(∫_^)((x2)/2⋅1/x*d(x))

5. Simplify the integral on the right side:

(∫_^)(x*ln(2*x)*d(x))=(x2)/2*ln(2*x)−(∫_^)(x/2*d(x))

6. Evaluate the remaining integral:

(∫_^)(x/2*d(x))=(x2)/4

7. Combine the terms and add the constant of integration C

Final Answer

(∫_^)(x*ln(2*x)*d(x))=(x2*ln(2*x))/2−(x2)/4+C

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