Evaluate the Integral

Problem

(∫_^)(ln(3*x)*d(x))

Solution

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

2. Assign variables for integration by parts by letting u=ln(3*x) and d(v)=d(x)

3. Differentiate u to find d(u)=1/(3*x)⋅3*d(x)=1/x*d(x)

4. Integrate d(v) to find v=x

5. Substitute these into the integration by parts formula:

(∫_^)(ln(3*x)*d(x))=x*ln(3*x)−(∫_^)(x⋅1/x*d(x))

6. Simplify the integrand in the second term:

(∫_^)(ln(3*x)*d(x))=x*ln(3*x)−(∫_^)(1*d(x))

7. Evaluate the remaining integral and add the constant of integration C

(∫_^)(ln(3*x)*d(x))=x*ln(3*x)−x+C

Final Answer

(∫_^)(ln(3*x)*d(x))=x*ln(3*x)−x+C

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