Can the Definite Integral of an Exponential Function be Expressed as a Product?

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In summary, the conversation discusses the integral $\int_0^\infty e^{-nx}x^{s-1}\,dx$ and shows that it is equal to $\frac{\prod(s-1)}{n^s}$. The conversation also touches on the definition of $\Pi$ and its relationship to factorials when working with natural numbers.
  • #1
Greg
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Show that $\int_0^\infty e^{-nx}x^{s-1}\,dx=\dfrac{\prod(s-1)}{n^s}$

I'm not familiar with the techniques that would be used to solve this.
 
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  • #2
I would likely begin by stating:

\(\displaystyle I(n,s-1)=\lim_{t\to\infty}\left(\int_0^t e^{-nx}x^{s-1}\,dx\right)\)

Okay, now, for the integral, I would use IBP, where:

\(\displaystyle u=x^{s-1}\,\therefore\,du=(s-1)x^{s-2}\,dx\)

\(\displaystyle dv=e^{-nx}\,dx\,\therefore\,v=-\frac{1}{n}e^{-nx}\)

And so we now have:

\(\displaystyle I(n,s-1)=\lim_{t\to\infty}\left(-\frac{1}{n}\left[x^{s-1}e^{-nx}\right]_0^t+\frac{s-1}{n}\int_0^t e^{-nx}x^{s-2}\,dx\right)\)

\(\displaystyle I(n,s-1)=\frac{s-1}{n}I(s-2,n)+\frac{1}{n}\lim_{t\to\infty}\left(\frac{x^{t-1}}{e^{nt}}\right)\)

We observe that the limit goes to zero (polynomial over an exponential) and we have:

\(\displaystyle I(n,s-1)=\frac{s-1}{n}I(n,s-2)\)

Now, if we repeat this process another $s-2$ times, we obtain:

\(\displaystyle I(n,s-1)=\frac{(s-1)!}{n^{s-1}}I(n,0)=\frac{(s-1)!}{n^{s-1}}\lim_{t\to\infty}\left(\int_0^t e^{-nx}\,dx\right)=\frac{(s-1)!}{n^s}\)
 
  • #3
Start by

$$\int_0^\infty e^{-sx}\,dx=\dfrac{1}{s}$$

Take the nth derivative with respect to $s$

$$\int_0^\infty e^{-sx}x ^n\,dx=\dfrac{n!}{s^{1+n}}$$
 
  • #4
Let's start with the definition of $\Pi$, which is:
$$\Pi(z)=\int_0^\infty e^{-t} t^z \,dt$$
Now we can substitute $z=s-1$ and $t=nx$.

Btw, if we start with the factorial, we can't "just" switch to $\Pi$ can we? :eek:
 
  • #5
I like Serena said:
Btw, if we start with the factorial, we can't "just" switch to $\Pi$ can we? :eek:

No we can't. This is under the assumption we work with natural numbers.
 

FAQ: Can the Definite Integral of an Exponential Function be Expressed as a Product?

What is a definite (improper) integral?

A definite (improper) integral is a mathematical concept used in calculus to find the area under a curve between two points. It is a way of summing up an infinite number of infinitely small values to find a more accurate total value.

How do you solve a definite (improper) integral?

To solve a definite (improper) integral, you must first identify the function being integrated and the limits of integration. Then, you can use various techniques, such as substitution, integration by parts, or the fundamental theorem of calculus, to evaluate the integral.

What is the difference between a definite integral and an indefinite integral?

The main difference between a definite integral and an indefinite integral is that a definite integral has specific limits of integration, while an indefinite integral does not. This means that a definite integral will result in a numerical value, while an indefinite integral will result in a function.

When should you use a definite (improper) integral?

A definite (improper) integral should be used when you need to find the exact area under a curve between two points, or when you need to find the total value of a continuous function over a specific interval. It is also used in various applications such as calculating work, distance, or volume.

Can a definite (improper) integral have a negative value?

Yes, a definite (improper) integral can have a negative value. This can happen when the function being integrated is negative over the given interval, or when the area under the curve is below the x-axis. In this case, the negative value represents a decrease in the quantity being measured.

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