Evaluate the Integral

Problem

(∫_^)(x*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 the formula using the LIATE rule, letting u=ln(x) and d(v)=x*d(x)

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

4. Integrate d(v) to find v=(x2)/2

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

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

6. Simplify the integral on the right side:

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

7. Evaluate the remaining integral:

(∫_^)(x/2*d(x))=(x2)/4

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

Final Answer

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

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