Evaluate the Integral

Problem

(∫_^)(1/(x*(ln(x))2)*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. Rewrite the integral in terms of u by replacing 1/x*d(x) with d(u) and ln(x) with u

(∫_^)(1/(u2)*d(u))

4. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)+C where n=−2

(∫_^)(u(−2)*d(u))=(u(−1))/(−1)+C

5. Simplify the resulting expression.

−1/u+C

6. Back-substitute the original variable x by replacing u with ln(x)

−1/ln(x)+C

Final Answer

(∫_^)(1/(x*(ln(x))2)*d(x))=−1/ln(x)+C

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