Rewriting an integral involving an exponential

In summary, in order to show the equation \int_0^{\infty}r^k e^{-a r} dr=\frac{k!}{a^{k+1}}, one can use the polynomial expansion of e^x and integration by parts. By letting the integral be I(k) and considering I(k-1), one can use the hint to do things by parts. This makes the proof much simpler.
  • #1
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Can someone help me show the following:

[tex]\int_0^{\infty}r^k e^{-a r} dr=\frac{k!}{a^{k+1}}[/tex]

I tried to use the polynomial expansion of e^x:

[tex]\sum_{n=0,1...} \frac{x^n}{n!}[/tex]

...but I get stuck pretty fast. Can someone give me a few hints?


Thanks!
 
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  • #2
Just do integration by parts and apply one more well known technique of proof (it might help if you let that integral be I(k), 'cos then what is I(k-1), and how does that link up with the hint to do things by parts?)
 
  • #3
I got it now! I wasn't really as difficult as it first seemed. Thanks alot! :-)
 

Related to Rewriting an integral involving an exponential

1. How do you rewrite an integral involving an exponential?

To rewrite an integral involving an exponential, you can use the substitution method, where you substitute a new variable for the exponential term and then solve the integral using that new variable. Another method is to use integration by parts, where you split the integral into two parts and apply the appropriate integration rules for each part.

2. What is the purpose of rewriting an integral involving an exponential?

The purpose of rewriting an integral involving an exponential is to make the integral easier to solve or to transform it into a different form that may be more useful for solving a particular problem.

3. Can rewriting an integral involving an exponential change its value?

Yes, rewriting an integral involving an exponential can change its value. This is because the substitution or integration by parts methods used to rewrite the integral may introduce new terms or change the limits of integration, which can affect the overall value of the integral.

4. Are there any special techniques for rewriting integrals involving complex exponentials?

Yes, for integrals involving complex exponentials, you can use the Euler's formula which states that e^(ix) = cos(x) + i sin(x). This can help simplify the integral and make it easier to solve using the substitution or integration by parts methods.

5. How can rewriting an integral involving an exponential be applied in real-world situations?

Rewriting integrals involving exponentials can be applied in many areas of science and engineering, such as in physics, chemistry, and economics. For example, in physics, integrals involving exponential functions can be used to model radioactive decay and population growth. In chemistry, they can be used to calculate reaction rates and concentrations. In economics, they can be used to analyze economic growth and inflation rates.

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