Evaluate the Integral

Problem

(∫_^)(ln(x)/(x4)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of a logarithmic function and an algebraic function, use integration by parts.

2. Assign variables for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=ln(x) and d(v)=x(−4)*d(x)

3. Differentiate u and integrate d(v) to find d(u) and v

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

v=(∫_^)(x(−4)*d(x))=(x(−3))/(−3)=−1/(3*x3)

4. Apply the integration by parts formula.

(∫_^)(ln(x)/(x4)*d(x))=−ln(x)/(3*x3)−(∫_^)(−1/(3*x3)⋅1/x*d(x))

5. Simplify the integral term.

(∫_^)(ln(x)/(x4)*d(x))=−ln(x)/(3*x3)+1/3*(∫_^)(x(−4)*d(x))

6. Evaluate the remaining integral.

(∫_^)(x(−4)*d(x))=−1/(3*x3)

7. Combine the terms and add the constant of integration C

(∫_^)(ln(x)/(x4)*d(x))=−ln(x)/(3*x3)+1/3*(−1/(3*x3))+C

8. Simplify the final expression.

(∫_^)(ln(x)/(x4)*d(x))=−ln(x)/(3*x3)−1/(9*x3)+C

Final Answer

(∫_^)(ln(x)/(x4)*d(x))=−(3*ln(x)+1)/(9*x3)+C

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