Evaluate the Integral

Problem

(∫_^)(((ln(x))5)/x*d(x))

Solution

1. Identify the structure of the integrand and notice that the derivative of ln(x) is 1/x which is present in the expression.

2. Apply the substitution method by letting u=ln(x)

3. Calculate the differential d(u) by differentiating u with respect to x resulting in d(u)=1/x*d(x)

4. Substitute u and d(u) into the original integral to rewrite it in terms of u

(∫_^)(u5*d(u))

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

(u6)/6+C

6. Back-substitute the original expression for u to get the final answer in terms of x

Final Answer

(∫_^)(((ln(x))5)/x*d(x))=((ln(x))6)/6+C

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