Evaluate the Integral

Problem

(∫_^)(x2*ln(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. Choose the components for integration by parts using the LIATE rule, setting u=ln(x) and d(v)=x2*d(x)

3. Differentiate u to find d(u)=1/x*d(x) and integrate d(v) to find v=(x3)/3

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

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

5. Simplify the integrand in the second term:

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

6. Integrate the remaining term:

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

7. Simplify the final expression:

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

Final Answer

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

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