Evaluate the Integral

Problem

(∫_^)(3/(x*ln(x))*d(x))

Solution

1. Identify the constant and move it outside the integral to simplify the expression.

(∫_^)(3/(x*ln(x))*d(x))=3*(∫_^)(1/(x*ln(x))*d(x))

2. Choose a substitution to simplify the denominator. Let u=ln(x)

u=ln(x)

3. Calculate the differential d(u) by differentiating u with respect to x

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

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

4. Substitute u and d(u) into the integral.

3*(∫_^)(1/u*d(u))

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

3*ln(u)+C

6. Back-substitute the original expression for u which is ln(x) to get the final result.

3*ln(ln(x))+C

Final Answer

(∫_^)(3/(x*ln(x))*d(x))=3*ln(ln(x))+C

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