Evaluate the Integral

Problem

(∫_1^3)(x2*ln(x)*d(x))

Solution

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

2. Assign the variables for integration by parts by letting u=ln(x) and d(v)=x2*d(x)

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

4. Apply the formula for integration by parts to the expression.

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

5. Simplify the integral on the right side.

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

6. Evaluate the remaining integral.

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

7. Apply the limits of integration from 1 to 3

[(x3)/3*ln(x)−(x3)/9]31

8. Substitute the upper limit x=3

3/3*ln(3)−3/9=9*ln(3)−3

9. Substitute the lower limit x=1

1/3*ln(1)−1/9=0−1/9=−1/9

10. Subtract the lower limit result from the upper limit result.

9*ln(3)−3−(−1/9)=9*ln(3)−3+1/9

11. Combine the constant terms.

9*ln(3)−27/9+1/9=9*ln(3)−26/9

Final Answer

(∫_1^3)(x2*ln(x)*d(x))=9*ln(3)−26/9

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