Evaluate the Integral

Problem

(∫_1^e)(ln(x)*d(x))

Solution

1. Identify the integration method required, which is 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)=d(x)

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

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

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

5. Simplify the integrand in the second term.

(∫_^)(ln(x)*d(x))=x*ln(x)−(∫_^)(1*d(x))

6. Evaluate the remaining integral.

(∫_^)(ln(x)*d(x))=x*ln(x)−x

7. Apply the limits of integration from 1 to e using the Fundamental Theorem of Calculus.

[x*ln(x)−x]e1

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

(e*ln(e)−e)−(1*ln(1)−1)

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

(e(1)−e)−(1*(0)−1)

10. Calculate the final result.

(e−e)−(0−1)=0+1=1

Final Answer

(∫_1^e)(ln(x)*d(x))=1

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