Evaluate the Integral

Problem

(∫_^)(ln(8*x)/x*d(x))

Solution

1. Identify the structure of the integrand and notice that the derivative of ln(8*x) is 1/x which suggests using the substitution method.

2. Substitute u=ln(8*x)

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=1/(8*x)⋅8

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

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

4. Rewrite the integral in terms of u

(∫_^)(u*d(u))

5. Integrate with respect to u using the power rule.

(u2)/2+C

6. Substitute back the original expression for u

((ln(8*x))2)/2+C

Final Answer

(∫_^)(ln(8*x)/x*d(x))=((ln(8*x))2)/2+C

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