Evaluate the Integral

Problem

(∫_^)(1/(x*ln(x4))*d(x))

Solution

1. Simplify the integrand using the logarithm power rule ln(ab)=b*ln(a)

(∫_^)(1/(x⋅4*ln(x))*d(x))

2. Factor out the constant coefficient from the integral.

1/4*(∫_^)(1/(x*ln(x))*d(x))

3. Identify a substitution to simplify the expression. Let u=ln(x)

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

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

1/4*(∫_^)(1/u*d(u))

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

1/4*ln(u)+C

6. Back-substitute the original variable u=ln(x) to find the final result.

1/4*ln(ln(x))+C

Final Answer

(∫_^)(1/(x*ln(x4))*d(x))=1/4*ln(ln(x))+C

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