Evaluate the Integral

Problem

(∫_^)(1/(x*ln(x))*d(x))

Solution

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

2. Substitute a new variable u for the logarithmic term.

u=ln(x)

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=1/x

d(u)=1/x*d(x)

4. Rewrite the integral in terms of u by substituting ln(x) and 1/x*d(x)

(∫_^)(1/u*d(u))

5. Integrate the expression with respect to u using the standard power rule for u(−1)

ln(u)+C

6. Back-substitute the original expression for u to return to the variable x

ln(ln(x))+C

Final Answer

(∫_^)(1/(x*ln(x))*d(x))=ln(ln(x))+C

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