Evaluate Improper Integral of Exponential Function

Evaluate the integral $\displaystyle \int_{1}^{\infty} e^{-2x} \, dx$.

Improper integrals extend the conventional concept of definite integrals to accommodate infinite limits of integration or integrands with infinite discontinuities. In solving improper integrals, one common strategy is to take the limit of a corresponding proper integral as the point of discontinuity approaches infinity or a specific point. For integrals like this one with an exponential function, we often evaluate the definite integral over a finite interval first, and then extend it to an infinite one by taking a limit.

This integral belongs to a class of problems often solved using straightforward applications of antiderivatives. The function e raised to the power of a negative constant times x is a common integrand in such problems, primarily involving exponential decay. Be mindful of the behavior of the exponential function as x approaches infinity; it tends towards zero, which influences the limit evaluation significantly.

Understanding improper integrals is crucial in various applications of mathematics, such as evaluating areas under curves when infinite integrals arise naturally—like in probability theory, quantum mechanics, and engineering.

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