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(a) Evaluate the integral $ \displaystyle \int_0^\infty e^{-x}\ dx $ for $ n = 0, 1, 2 $, and $ 3 $.(b) Guess the value of $ \displaystyle \int_0^\infty x^n e^{-x}\ dx $ when $ n $ is an arbitrary positive integer.(c) Prove your guess using mathematical induction.

a. $1,1,2,6$b. $\int_{0}^{\infty} x^{n} e^{-x} \mathrm{d} x=n !$c. $n !$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

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In mathematics, integratio…

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(a) Use integration by par…

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I'm into girl from zero to a into negative ads. Arms as to is Howard. Yes. Yes, we assume. What is that? Okay, the result is K factorial. So here, when a goes to infinity this's k factorial on k factorial times Keep us one is K Paswan factorial on the first part. This's goes to Cyril Want a costly infinity? You can also use a big ass road proof This is zero. So the result is a class one by induction. So our guys, this is true. We have proved our guests.

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