Evaluate the Integral

Problem

(∫_^)(((ln(x))6)/x*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of ln(x) 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 solving for d(x) resulting in d(x)=x*d(u)

4. Substitute these values into the original integral to eliminate x

(∫_^)((u6)/x⋅x*d(u))

5. Simplify the expression by canceling the x terms.

(∫_^)(u6*d(u))

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

(u7)/7+C

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

Final Answer

(∫_^)(((ln(x))6)/x*d(x))=((ln(x))7)/7+C

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