Evaluate the definite integral.
[tex]\int_1^{e^3} \frac{d x}{x \sqrt{\ln x}}[/tex]
Answer (1)
Perform a u-substitution with u = ln x , so d u = x 1 d x .
Change the limits of integration: from x = 1 to u = 0 , and from x = e 3 to u = 3 .
Rewrite the integral as ∫ 0 3 u − 1/2 d u .
Evaluate the integral to get 2 3 .
Explanation
Problem Setup We are asked to evaluate the definite integral ∫ 1 e 3 x ln x d x
U-Substitution Let's use the substitution method. Let u = ln x . Then, the derivative of u with respect to x is d x d u = x 1 , which means d u = x 1 d x .
Changing Limits of Integration Now we need to change the limits of integration. When x = 1 , u = ln ( 1 ) = 0 . When x = e 3 , u = ln ( e 3 ) = 3 . So, the new limits of integration are from 0 to 3 .
Rewriting the Integral We can rewrite the integral in terms of u :
∫ 0 3 u 1 d u = ∫ 0 3 u − 1/2 d u
Evaluating the Integral Now, let's evaluate the integral: ∫ 0 3 u − 1/2 d u = [ 1/2 u 1/2 ] 0 3 = [ 2 u ] 0 3 = 2 3 − 2 0 = 2 3
Final Answer Therefore, the value of the definite integral is 2 3 .
Examples
Imagine you're analyzing the flow of data in a computer network where the rate of data transfer decreases logarithmically with distance. Evaluating an integral like this can help you determine the total amount of data transferred over a certain range, which is crucial for optimizing network performance and ensuring efficient communication across different nodes.
