Evaluate the Integral

Problem

(∫_^)(8*x2*ln(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 substitution by letting u=ln(x) and d(v)=8*x2*d(x)

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

4. Integrate d(v) to find v=(∫_^)(8*x2*d(x))=(8*x3)/3

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

(∫_^)(8*x2*ln(x)*d(x))=ln(x)⋅(8*x3)/3−(∫_^)((8*x3)/3⋅1/x*d(x))

6. Simplify the integral on the right side:

(∫_^)(8*x2*ln(x)*d(x))=(8*x3*ln(x))/3−(∫_^)((8*x2)/3*d(x))

7. Evaluate the remaining integral:

(∫_^)((8*x2)/3*d(x))=8/3⋅(x3)/3=(8*x3)/9

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

Final Answer

(∫_^)(8*x2*ln(x)*d(x))=(8*x3*ln(x))/3−(8*x3)/9+C

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