Evaluate the Integral

Problem

(∫_^)(((ln(x))9)/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 substitution by letting u=ln(x)

3. Calculate the differential d(u) by differentiating u with respect to x

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

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

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

(∫_^)(u9*d(u))

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

(u10)/10+C

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

((ln(x))10)/10+C

Final Answer

(∫_^)(((ln(x))9)/x*d(x))=((ln(x))10)/10+C

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