Evaluate the Integral

Problem

(∫_^)(x7*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)=x7*d(x)

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

4. Integrate d(v) to find v=(x8)/8

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

(∫_^)(x7*ln(x)*d(x))=ln(x)⋅(x8)/8−(∫_^)((x8)/8⋅1/x*d(x))

6. Simplify the integral on the right side.

(∫_^)(x7*ln(x)*d(x))=(x8*ln(x))/8−(∫_^)((x7)/8*d(x))

7. Evaluate the remaining integral.

(∫_^)((x7)/8*d(x))=1/8⋅(x8)/8=(x8)/64

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

Final Answer

(∫_^)(x7*ln(x)*d(x))=(x8*ln(x))/8−(x8)/64+C

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