Evaluate the Integral

Problem

(∫_^)(x5*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 parts for substitution by letting u=ln(x) and d(v)=x5*d(x)

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

4. Integrate d(v) to find v=(x6)/6

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

(∫_^)(x5*ln(x)*d(x))=ln(x)⋅(x6)/6−(∫_^)((x6)/6⋅1/x*d(x))

6. Simplify the integral on the right side.

(∫_^)(x5*ln(x)*d(x))=(x6*ln(x))/6−(∫_^)((x5)/6*d(x))

7. Evaluate the remaining integral.

(∫_^)((x5)/6*d(x))=1/6⋅(x6)/6=(x6)/36

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

Final Answer

(∫_^)(x5*ln(x)*d(x))=(x6*ln(x))/6−(x6)/36+C

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