7–58. Improper integrals Evaluate the following integrals or state that they diverge.
10. ∫ (from 0 to ∞) e⁻²ˣ dx

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Identify the type of improper integral: Since the upper limit of integration is infinity, this is an improper integral of the form \(\int_0^{\infty} e^{-2x} \, dx\).

Rewrite the integral as a limit to handle the infinite upper bound: Express the integral as \(\lim_{t \to \infty} \int_0^t e^{-2x} \, dx\).

Find the antiderivative of the integrand \(e^{-2x}\): Recall that the integral of \(e^{ax}\) with respect to \(x\) is \(\frac{1}{a} e^{ax}\), so here the antiderivative is \(-\frac{1}{2} e^{-2x}\).

Evaluate the definite integral from 0 to \(t\) using the antiderivative: Substitute the limits to get \(\left[-\frac{1}{2} e^{-2x} \right]_0^t = -\frac{1}{2} e^{-2t} + \frac{1}{2}\).

Take the limit as \(t\) approaches infinity: Calculate \(\lim_{t \to \infty} \left(-\frac{1}{2} e^{-2t} + \frac{1}{2}\right)\) to determine if the integral converges or diverges.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, limits are used to handle the infinite bounds or singularities, determining if the integral converges to a finite value or diverges.

Exponential Decay Functions

Exponential decay functions, like e^(-2x), decrease rapidly as x increases. Their integrals over infinite intervals often converge because the function approaches zero fast enough, making the area under the curve finite.

Evaluating Definite Integrals Using Limits

When integrating over infinite limits, the definite integral is expressed as a limit of integrals with finite bounds. This approach allows calculation of the integral's value by taking the limit as the bound approaches infinity.

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