Evaluate the Integral

Problem

(∫_^)(1/(x*ln(x))*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the derivative of ln(x) is 1/x which is present in the integrand.

2. Substitute u=ln(x)

3. Differentiate u to find d(u)=1/x*d(x)

4. Rewrite the integral in terms of u by substituting the expressions from the previous steps.

(∫_^)(1/(u2)*d(u))

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

−u(−1)+C

6. Simplify the expression to a fraction.

−1/u+C

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

Final Answer

(∫_^)(1/(x*ln(x))*d(x))=−1/ln(x)+C

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