Evaluate the Integral

Problem

(∫_^)(ln(x)/(x5)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of a logarithmic function and a power function, use integration by parts with the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign variables for integration by parts. Let u=ln(x) and d(v)=x(−5)*d(x)

3. Differentiate u and integrate d(v)

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

v=(∫_^)(x(−5)*d(x))=(x(−4))/(−4)=−1/(4*x4)

4. Apply the integration by parts formula.

(∫_^)(ln(x)/(x5)*d(x))=−ln(x)/(4*x4)−(∫_^)(−1/(4*x4)⋅1/x*d(x))

5. Simplify the integral term.

(∫_^)(ln(x)/(x5)*d(x))=−ln(x)/(4*x4)+1/4*(∫_^)(x(−5)*d(x))

6. Evaluate the remaining integral.

(∫_^)(ln(x)/(x5)*d(x))=−ln(x)/(4*x4)+1/4*(−1/(4*x4))+C

7. Combine the terms and factor out common constants.

(∫_^)(ln(x)/(x5)*d(x))=−ln(x)/(4*x4)−1/(16*x4)+C

Final Answer

(∫_^)(ln(x)/(x5)*d(x))=−(4*ln(x)+1)/(16*x4)+C

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