An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
We are given the integral ∫ x 13 e x 14 d x .
Apply the substitution u = x 14 , which gives d u = 14 x 13 d x .
Rewrite the integral as ∫ 14 1 e u d u .
Evaluate the integral and substitute back to get 14 1 e ( x 14 ) + C .
Explanation
Problem Setup and Substitution We are given the integral ∫ x 13 e x 14 d x and asked to evaluate it. We are also given the substitution u = x 14 . Let's proceed with this substitution.
Finding du and Substituting First, we find the derivative of u with respect to x :
d x d u = 14 x 13 This implies that d x = 14 x 13 d u Now, we substitute u and d x into the original integral: ∫ x 13 e x 14 d x = ∫ x 13 e u 14 x 13 d u = ∫ 14 1 e u d u So, the integral in terms of u is ∫ 14 1 e u d u .
Evaluating the Integral and Substituting Back Next, we evaluate the integral with respect to u :
∫ 14 1 e u d u = 14 1 e u + C Finally, we substitute back u = x 14 to express the result in terms of x :
14 1 e x 14 + C Thus, the evaluated integral is 14 1 e ( x 14 ) .
Examples
Imagine you're analyzing the spread of a computer virus. The rate of infection might be modeled by a function similar to the one in this problem, where the exponent represents the complexity of the system. Evaluating such integrals helps predict the total number of infected computers over time, allowing for timely intervention and prevention strategies. Understanding substitution techniques is crucial for modeling and controlling complex systems in various fields.
The integral ∫ x 13 e x 14 d x can be evaluated by using the substitution u = x 14 , leading to the result 14 1 e ( x 14 ) + C . After changing variables, the simplified integral can be evaluated directly, and then we convert back to the original variable. Therefore, the answer is 14 1 e ( x 14 ) + C .
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