Evaluate the Integral

Problem

(∫_^)(((ln(x))2)/(x3)*d(x))

Solution

1. Identify the integration method as integration by parts, using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=(ln(x))2 and d(v)=x(−3)*d(x)

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

d(u)=(2*ln(x))/x*d(x)

v=(∫_^)(x(−3)*d(x))=(x(−2))/(−2)=−1/(2*x2)

3. Apply the integration by parts formula for the first time.

(∫_^)(((ln(x))2)/(x3)*d(x))=−((ln(x))2)/(2*x2)−(∫_^)(−1/(2*x2)⋅(2*ln(x))/x*d(x))

4. Simplify the resulting integral.

(∫_^)(((ln(x))2)/(x3)*d(x))=−((ln(x))2)/(2*x2)+(∫_^)(ln(x)/(x3)*d(x))

5. Apply integration by parts again to the new integral (∫_^)(ln(x)/(x3)*d(x)) Let u=ln(x) and d(v)=x(−3)*d(x)

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

v=−1/(2*x2)

6. Evaluate the second integration by parts.

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)−(∫_^)(−1/(2*x2)⋅1/x*d(x))

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)+1/2*(∫_^)(x(−3)*d(x))

7. Compute the final integral and combine all terms.

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)+1/2*(−1/(2*x2))

(∫_^)(ln(x)/(x3)*d(x))=−ln(x)/(2*x2)−1/(4*x2)

8. Substitute this result back into the expression from step 4 and add the constant of integration C

(∫_^)(((ln(x))2)/(x3)*d(x))=−((ln(x))2)/(2*x2)−ln(x)/(2*x2)−1/(4*x2)+C

Final Answer

(∫_^)(((ln(x))2)/(x3)*d(x))=−(2*(ln(x))2+2*ln(x)+1)/(4*x2)+C

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