Evaluate the Integral

Problem

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

3. Rewrite the integral in terms of u by replacing (ln(x))3 with u3 and 1/x*d(x) with d(u)

(∫_^)(u3*d(u))

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

(u4)/4+C

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

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

Final Answer

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

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