Evaluate the Integral

Problem

(∫_^)(1/(x*(ln(x))4)*d(x))

Solution

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

2. Substitute u=ln(x) which implies that the derivative is d(u)/d(x)=1/x

3. Rewrite the differential d(x) in terms of d(u) by using d(u)=1/x*d(x)

4. Transform the integral into the uvariable to get (∫_^)(1/(u4)*d(u))

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

6. Simplify the resulting expression to −1/(3*u3)+C

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

Final Answer

(∫_^)(1/(x*(ln(x))4)*d(x))=−1/(3*(ln(x))3)+C

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