Evaluate the Integral

Problem

(∫_1^2)(x*ln(x)*d(x))

Solution

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

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

3. Differentiate u to find d(u)=1/x*d(x) and integrate d(v) to find v=(x2)/2

4. Apply the formula for integration by parts to the definite integral.

(∫_1^2)(x*ln(x)*d(x))=[(x2)/2*ln(x)]21−(∫_1^2)((x2)/2⋅1/x*d(x))

5. Simplify the integral term on the right side.

(∫_1^2)(x/2*d(x))

6. Evaluate the simplified integral.

[(x2)/4]21

7. Combine the evaluated parts and substitute the limits of integration.

(2/2*ln(2)−1/2*ln(1))−(2/4−1/4)

8. Simplify the numerical values, noting that ln(1)=0

(2*ln(2)−0)−(1−1/4)

9. Calculate the final result.

2*ln(2)−3/4

Final Answer

(∫_1^2)(x*ln(x)*d(x))=2*ln(2)−3/4

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