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Improper Integrals: Example 9: Comparison Theorem

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First published at 23:35 UTC on May 28th, 2019.

#mathematics
#calculus
#science

Math Easy Solutions

MES

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In this video I go over another example on improper integrals and this time show how to determine if the integral of the function e^(-x^2) from x = 0 to x approaches infinity is convergent or divergent. This function is very interesting in that the integral is not an elementary function so that we can't directly solve for the integral of that function. Instead we have to use the Comparison Theorem for improper integrals to instead compare it with a function that we know how to evaluate the integral of. That particular function that I go over in this example is the function e^(-x) and show how since the integral of this function is convergent, and since e^(-x) is always greater than e^(-x^2) for x is greater than 1, then the integral of the function e^(-x^2) is also convergent. Thus we can determine if an integral converges without having to evaluate it directly.

In later videos I will how to solve for the exact value of this integral, which is very useful for probability theory, which I will cover later too.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhqVSDt3w1CfcyPxvlQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/improper-integrals-example-9-comparison-theorem

Related Videos:

Comparison Theorem for Type 1 Improper Integrals: https://youtu.be/SswR3u6rurU
Can We Integrate All Continuous Functions?: http://youtu.be/OFEDLJYqYps
Improper Integrals: Type 2: Discontinuous Integrands: https://youtu.be/livlGUBuM54
Improper Integrals: Type 1: Infinite Intervals: https://youtu.be/9ZlsCy3wdZU
Improper Integrals: Introduction: https://youtu.be/ud44W-0jC-c .

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