Evaluate the Integral

Problem

(∫_1^e)(x2*ln(x)*d(x))

Solution

1. Identify the method of integration by parts, using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=ln(x) and d(v)=x2*d(x)

2. Differentiate u to find d(u)=1/x*d(x) and integrate d(v) to find v=(x3)/3

3. Apply the formula for integration by parts to the indefinite integral.

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

4. Simplify the integral on the right side.

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

5. Evaluate the remaining integral.

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

6. Apply the limits of integration from 1 to e

[(x3)/3*ln(x)−(x3)/9]e1

7. Substitute the upper limit e and the lower limit 1 into the expression.

((e3)/3*ln(e)−(e3)/9)−(1/3*ln(1)−1/9)

8. Simplify using the values ln(e)=1 and ln(1)=0

((e3)/3−(e3)/9)−(0−1/9)

9. Combine the terms to find the final value.

(3*e3−e3)/9+1/9

(2*e3+1)/9

Final Answer

(∫_1^e)(x2*ln(x)*d(x))=(2*e3+1)/9

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