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Determine whether the integral \int_0^\infty x^2 e^{-x} \,dx converges or diverges. If...

Question:

Determine whether the integral {eq}\int_0^\infty x^2 e^{-x} \,dx {/eq} converges or diverges. If convergent, evaluate it.

Improper Integrals; Convergence:

We recall that

{eq}\int_0^\infty x^2 e^{-x} \,dx=\lim\limits_{k\to\infty} \int_0^{k} x^2 e^{-x} \,dx . {/eq}

To evaluate,

{eq}\int_0^{k} x^2 e^{-x} \,dx, {/eq}

we'll integrate by parts twice.

To find the limit we'll use the fact that the exponential {eq}e^x {/eq} grows faster than any polynomial as {eq}n\to\infty. {/eq}

Alternatively, to find the limit we could use L'Hospitals' Rule twice.

Answer and Explanation:

To analyze the improper integral {eq}I= \int_0^\infty x^2 e^{-x} \,dx {/eq}

we recall that,

{eq}I=\lim\limits_{k\to\infty} \int_0^{k} x^2 e^{-x}...

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How to Solve Improper Integrals

from Math 104: Calculus

Chapter 13 / Lesson 12
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