Evaluate the Integral

Problem

(∫_e^e2)(1/(x*ln(x))*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of ln(x) is 1/x which 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=e u=ln(e)=1 when x=e2 u=ln(e2)=2

4. Rewrite the integral in terms of u using the new limits and the substitution.

(∫_1^2)(1/u*d(u))

5. Integrate the expression using the rule (∫_^)(1/u*d(u))=ln(u)

[ln(u)]21

6. Evaluate the definite integral by subtracting the value at the lower limit from the value at the upper limit.

ln(2)−ln(1)

7. Simplify the result using the fact that ln(1)=0

ln(2)−0=ln(2)

Final Answer

(∫_e^e2)(1/(x*ln(x))*d(x))=ln(2)

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