[arctan e^(x)]_(0)^(oo)=

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Recognize Problem: Recognize that the problem is asking for the evaluation of a definite integral of the function arctan(e^x) from x = 0 to x = infinity.Set Up Integral: Set up the integral for the given function. The integral we need to evaluate is ∫ from 0 to ∞ of arctan(e^x) dx.Behavior Analysis: Consider the behavior of the function as x approaches infinity. As x approaches infinity, e^x also approaches infinity, and arctan(e^x) approaches π/2 because arctan(∞) = π/2.Evaluate at Limits: Consider the behavior of the function as x approaches 0. As x approaches 0, e^x approaches 1, and arctan(e^x) approaches arctan(1) which is π/4 because arctan(1) = π/4.No Elementary Antiderivative: Evaluate the integral using the limits of integration. The integral of arctan(e^x) from 0 to ∞ is the limit as t approaches ∞ of the integral of arctan(e^x) from 0 to t minus the integral from 0 to 0, which is 0. So, we need to evaluate the limit as t approaches ∞ of ∫ from 0 to t of arctan(e^x) dx.Use Improper Integral: Realize that the integral does not have an elementary antiderivative. The function arctan(e^x) does not have an elementary antiderivative, so we cannot evaluate the integral using basic integration techniques.Apply Limits: Use improper integral evaluation techniques. Since we cannot find an antiderivative, we must evaluate the integral as an improper integral by taking the limit as t approaches infinity of the integral from 0 to t of arctan(e^x) dx.Realization of Limitation: Attempt to apply limits to evaluate the improper integral. We need to find the limit as t approaches infinity of the integral from 0 to t of arctan(e^x) dx. However, without an antiderivative, we cannot directly evaluate this limit.Realize that the problem cannot be solved using standard calculus techniques. Without an antiderivative or a known convergence theorem that applies, we cannot evaluate the integral from 0 to infinity of arctan(e^x) dx using the information given.

$\left[\arctan e^{x}\right]_{0}^{\infty}$ =

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[arctan⁡ex]0∞\left[\arctan e^{x}\right]_{0}^{\infty}[arctanex]0∞​=

$\left[\arctan e^{x}\right]_{0}^{\infty}$ =