Evaluate the Integral

Problem

(∫_1^8)(ln(x)/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 d(u)=1/x*d(x)

3. Change the limits of integration to correspond with u when x=1 u=ln(1)=0 when x=8 u=ln(8)

4. Rewrite the integral in terms of u

(∫_0^ln(8))(u*d(u))

5. Integrate with respect to u using the power rule:

[(u2)/2]ln(8)0

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

((ln(8))2)/2−0/2

7. Simplify the result:

((ln(8))2)/2

Final Answer

(∫_1^8)(ln(x)/x*d(x))=((ln(8))2)/2

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