Evaluate the Integral

Problem

(∫_1^2)(((ln(x))2)/(x3)*d(x))

Solution

1. Identify the integration method as integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Set u=(ln(x))2 and d(v)=x(−3)*d(x)

3. Differentiate u to find d(u)=2*ln(x)⋅1/x*d(x) and integrate d(v) to find v=−1/(2*x2)

4. Apply the integration by parts formula:

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

5. Simplify the new integral:

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

6. Apply integration by parts again for the second integral, setting u=ln(x) and d(v)=x(−3)*d(x) which gives d(u)=1/x*d(x) and v=−1/(2*x2)

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

7. Evaluate the remaining integral:

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

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

8. Combine all parts to find the indefinite integral:

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

9. Evaluate the definite integral from 1 to 2

(∫_1^2)(((ln(x))2)/(x3)*d(x))=[−(2*(ln(x))2+2*ln(x)+1)/(4*x2)]21

10. Substitute the upper and lower limits:

(−(2*(ln(2))2+2*ln(2)+1)/(4*(2)2))−(−(2*(ln(1))2+2*ln(1)+1)/(4*(1)2))

11. Simplify using ln(1)=0

−(2*(ln(2))2+2*ln(2)+1)/16+1/4

12. Combine terms over a common denominator:

(−2*(ln(2))2−2*ln(2)−1+4)/16

Final Answer

(∫_1^2)(((ln(x))2)/(x3)*d(x))=(3−2*ln(2)−2*(ln(2))2)/16

Want more problems? Check here!