Evaluate the Integral

Problem

(∫_^)(x2*ln(3*x)*d(x))

Solution

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

2. Choose the components for integration by parts by letting u=ln(3*x) and d(v)=x2*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=(x3)/3

5. Substitute these components into the integration by parts formula:

(∫_^)(x2*ln(3*x)*d(x))=ln(3*x)⋅(x3)/3−(∫_^)((x3)/3⋅1/x*d(x))

6. Simplify the integral on the right side:

(∫_^)(x2*ln(3*x)*d(x))=(x3*ln(3*x))/3−(∫_^)((x2)/3*d(x))

7. Evaluate the remaining integral:

(∫_^)((x2)/3*d(x))=1/3⋅(x3)/3=(x3)/9

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

Final Answer

(∫_^)(x2*ln(3*x)*d(x))=(x3*ln(3*x))/3−(x3)/9+C

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