What is the Pattern in the Integration of (x^n)(e^x) dx?

In summary: In = xnex - n(xn-1ex - (n-1)In-2)In = xnex - nxn-1ex + n(n-1)In-2so in general In = xnex - nxn-1ex + n(n-1)In-2which is the same as your result. Thus, the pattern is proven. In summary, the conversation discusses the topic of integration by parts and a pattern that is observed when integrating (x^n)(e^x). The pattern is proven using the concepts of derivatives and generating functions.
  • #1
zketrouble
47
0
so I just introduced myself to the concept of integration by parts a few days ago, and today for practice I decided to try to redo one of the examples given in a youtube video without the assistance of youtube. That problem was int. (x^2)(e^x) dx, for which the answer is (e^x)(x^2 -2x + 2). So then I tried to integrate (x^3)(e^x), (x^4)(e^x), and so on, a drew the results in a table:

y = int. (x^n)(e^x)

n=0; y = e^x
n=1; y = e^x(x-1)
n=2; y = e^x(x^2 - 2x + 2)
n=3; y = e^x(x^3 - 3x^2 + 6x -6)
n=4; y = e^x(x^4 - 4x^3 + 12x^2 - 24x + 24)
n=5; y = e^x(x^5 - 5x^4 + 20x^3 - 60x^2 + 120x -120)

I noticed a pattern, and I'm curious as to how I would formally prove this pattern:

(x^n)(e^x) dx =
(e^x){x^n - nx^(n-1) + n(n-1)x^(n-2) - n(n-1)(n-2)x^(n-3) + n(n-1)(n-2)(n-3)x^(n-4)...-+-+-+...}

It is very likely that the procedures necessary to prove this are way beyond my current calculus skills, but I'm curious anyway to see what the folks here at PF can come up with to prove/disprove my assumption that the above statement is true.
 
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  • #2
It's not calculus needed to prove the general 'n' case, but mathematical induction.

Yay, 10^2^2 posts :biggrin:
 
  • #3
silly me...forgot all of the + c every time. :p
 
  • #4
>n=5; y = e^x(x^5 - 5x^4 + 20x^3 - 60x^2 + 120x -120)
>I noticed a pattern, and I'm curious as to how I would formally prove this pattern:

You don't need induction to prove your pattern. If you take the derivative of one of your examples

d/dx [e^x(x^5 - 5x^4 + 20x^3 - 60x^2 + 120x -120)] =
e^x(x^5 - 5x^4 + 20x^3 - 60x^2 + 120x -120)
+e^x(5x^4 - 20x^3 + 60x^2 -120x + 120) = e^x x^5

you can see that every term but the first vanishes. Since the expression produces the derivative you wanted (e^x x^5), it must be the correct antiderivative (though any constant can be added of course). Just do the same thing with your formula and you've proven it.
 
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  • #5
An advanced concept that is very useful here is that of a generating function. Consider

[tex]F(a) = \int e^{ax} dx= \frac{e^{ax}}{a} + c.[/tex]

We can compute

[tex]\frac{d^nF(a)}{da^n} = \int x^n e^{ax} dx,[/tex]

so that

[tex]I_n = \int x^n e^x dx = \left. \frac{d^nF(a)}{da^n}\right|_{a=1}. [/tex]

Therefore we can generate any of the [tex]I_n[/tex] just by computing derivatives of [tex]F(a)[/tex]. These derivatives are just the same derivatives that you compute when integrating by parts.
 
  • #6
your result is correct

let In be your integral; if you integrate by parts In = xnex - nIn-1
 

1. What is the general pattern for integrating (x^n)(e^x)?

The general pattern for integrating (x^n)(e^x) is (x^n)(e^x) = (x^(n+1))(e^x) - n∫(x^(n-1))(e^x) dx.

2. How do I handle the integral of (x^n)(e^x) when n is a negative number?

If n is a negative number, you can use the power rule for integration to rewrite the integral as (-1)^n∫(x^n)(e^x) dx. Then, you can use the general pattern for integrating (x^n)(e^x) as mentioned in question 1.

3. Can (x^n)(e^x) be integrated by parts?

Yes, (x^n)(e^x) can be integrated by parts. The formula for integration by parts is ∫u dv = uv - ∫v du. You can choose u and dv such that after applying the formula, you can simplify the integral into a form that is easier to integrate.

4. Is there a specific substitution that can be used to simplify the integral of (x^n)(e^x)?

Yes, there is a substitution that can be used to simplify the integral of (x^n)(e^x). If n is a positive integer, you can use the substitution u = x^n to transform the integral into ∫(u^(1/n))(e^(u^(1/n))) du, which can then be integrated using integration by parts.

5. Can the integral of (x^n)(e^x) be evaluated using a computer program?

Yes, the integral of (x^n)(e^x) can be evaluated using a computer program. Most programming languages have built-in functions or libraries for numerical integration, which can be used to calculate the value of the integral. However, it is important to note that such numerical methods may not always provide an exact solution and may only give an approximation.

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