Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to ∞ of (x² * e^(−x)) dx

검증된 단계별 안내

Recognize that the integral \( \int_0^{\infty} x^2 e^{-x} \, dx \) is an improper integral because the upper limit is infinity. This means we need to evaluate it as a limit: \( \lim_{t \to \infty} \int_0^t x^2 e^{-x} \, dx \).

Set up the integral with the limit: \( \lim_{t \to \infty} \int_0^t x^2 e^{-x} \, dx \). We will first find the antiderivative of \( x^2 e^{-x} \) and then apply the limits.

Use integration by parts to find the antiderivative. Let \( u = x^2 \) and \( dv = e^{-x} dx \). Then, compute \( du = 2x dx \) and \( v = -e^{-x} \). Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \).

After the first integration by parts, you will get an integral involving \( x e^{-x} \). Apply integration by parts again on this new integral, letting \( u = x \) and \( dv = e^{-x} dx \), and repeat the process to fully evaluate the integral.

Once you have the antiderivative, evaluate it at the limits 0 and \( t \), then take the limit as \( t \to \infty \). This will give you the value of the improper integral.

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주요 개념

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, the integral is expressed as a limit where the bound approaches infinity or the point of discontinuity. Convergence or divergence is determined by the existence of this limit.

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation, used to integrate products of functions. It transforms the integral of u dv into uv minus the integral of v du, simplifying complex integrals like x² e^(−x). Choosing u and dv wisely is key to success.

Gamma Function and Factorials

The Gamma function generalizes factorials to non-integer values and is defined as an improper integral involving x^(n) e^(−x). Recognizing integrals of the form ∫₀^∞ x^n e^(−x) dx as Gamma functions helps evaluate them quickly, where Γ(n+1) = n! for positive integers n.

Evaluate the improper integrals in Exercises 53–62.∫ from 0 to ∞ of (x² * e^(−x)) dx