Evaluate the Integral

Problem

(∫_1^e)(ln(x)/x*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of ln(x) is present in the integrand.

2. Substitute u=ln(x) which implies that the differential d(u)=1/x*d(x)

3. Change the limits of integration to correspond with the new variable u when x=1 u=ln(1)=0 when x=e u=ln(e)=1

4. Rewrite the integral in terms of u

(∫_0^1)(u*d(u))

5. Apply the power rule for integration, which states that (∫_^)(un*d(u))=(u(n+1))/(n+1)

[(u2)/2]10

6. Evaluate the expression at the upper and lower limits:

1/2−0/2=1/2

Final Answer

(∫_1^e)(ln(x)/x*d(x))=1/2

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