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-x2 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-x2 for x is greater than 1, then the integral of the function e-x2 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.
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Improper Integrals: Example 9
Show that the following integral is convergent:
Solution
As discussed in my earlier video titled "Can we integrate all continuous functions?", this function has no direct solution because it is not an elementary function.
So let's compare the integrand with one that we know how to integrate, y = e-x,
Important Notes
In this example, we were able to show that the integral:
is convergent without computing its value.
In probability theory it is important to know the exact value of this integral, as I will show in my later videos so stay tuned for those!
Using multivariable calculus, it can be shown that the exact value is:
The table below, which comprise of computer generated values of this integral, illustrate this improper integral quite well. From the table the integral converges very fast to the exact value of √π/2. This is because e-x2→0 very fast as x → ∞.
