Evaluate the Integral

Problem

(∫_1^7)(((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)) Let u=(ln(x))2 and d(v)=x(−3)*d(x)

2. Differentiate u to find d(u)=2*ln(x)⋅1/x*d(x)=(2*ln(x))/x*d(x)

3. Integrate d(v) to find v=(x(−2))/(−2)=−1/(2*x2)

4. 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))

5. Simplify the resulting 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 new integral (∫_^)(ln(x)/(x3)*d(x)) Let u=ln(x) and d(v)=x(−3)*d(x) Then 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. Simplify and evaluate the remaining integral.

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

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

8. Combine all parts to find the general antiderivative.

(∫_^)(((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 7 by substituting the upper and lower limits.

[−(2*(ln(x))2+2*ln(x)+1)/(4*x2)]71

10. Calculate the value at x=7 and x=1

(−(2*(ln(7))2+2*ln(7)+1)/(4*(7)2))−(−(2*(ln(1))2+2*ln(1)+1)/(4*(1)2))

(−(2*(ln(7))2+2*ln(7)+1)/196)−(−1/4)

11. Simplify the final expression using a common denominator.

49/196−(2*(ln(7))2+2*ln(7)+1)/196

(48−2*ln(7)−2*(ln(7))2)/196

(24−ln(7)−(ln(7))2)/98

Final Answer

(∫_1^7)(((ln(x))2)/(x3)*d(x))=(24−ln(7)−(ln(7))2)/98

Want more problems? Check here!