Evaluate the Integral

Problem

(∫_^)(x8*ln(x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the parts by letting u=ln(x) and d(v)=x8*d(x)

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

4. Integrate d(v) to find v=(x9)/9

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

(∫_^)(x8*ln(x)*d(x))=ln(x)⋅(x9)/9−(∫_^)((x9)/9⋅1/x*d(x))

6. Simplify the integral on the right:

(∫_^)(x8*ln(x)*d(x))=(x9*ln(x))/9−(∫_^)((x8)/9*d(x))

7. Evaluate the remaining integral:

(∫_^)(x8*ln(x)*d(x))=(x9*ln(x))/9−1/9⋅(x9)/9+C

8. Simplify the final expression:

(∫_^)(x8*ln(x)*d(x))=(x9*ln(x))/9−(x9)/81+C

Final Answer

(∫_^)(x8*ln(x)*d(x))=(x9*ln(x))/9−(x9)/81+C

Want more problems? Check here!