Evaluate the Integral

Problem

(∫_^)(ln(x6)/x*d(x))

Solution

1. Apply logarithm properties to simplify the integrand by moving the exponent outside the natural log.

ln(x6)=6*ln(x)

2. Rewrite the integral with the constant factor moved outside the integral sign.

6*(∫_^)(ln(x)/x*d(x))

3. Identify a substitution by letting u=ln(x)

u=ln(x)

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

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

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

5. Substitute u and d(u) into the integral.

6*(∫_^)(u*d(u))

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

6⋅(u2)/2+C

3*u2+C

7. Substitute back the original expression for u to get the final result in terms of x

3*(ln(x))2+C

Final Answer

(∫_^)(ln(x6)/x*d(x))=3*(ln(x))2+C

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