I with finding the intergral of (x^n)(e^x)

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In summary, for finding the integral of (x^n)(e^x), you can use integration by parts and the recurrence relation to get the general formula of x^n*e^x - n* integral of (x^(n-1))(e^x). This can also be used to find the integral for e^(-x) and is a definition for the gamma function.
  • #1
minase
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I need help with finding the intergral of (x^n)(e^x). I already did for 1,2 and 3. I found out that x^n*e^x times the intergral of (x^n-1)e^x. Is there anyway I can genralize for all n s without having an integral.
 
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  • #2
Post this in the homework section "Calculus and Beyond" and you are much more likely to get an answer.
 
  • #3
Easy, use integration by part, show the following relationship
[tex] \int x^n e^x dx = x^ne^x -n\int x^{n-1}e^x dx [/tex]

Then use the recurrence relation on [tex] \int x^{n-1} e^x dx [/tex].
You will get
[tex] \int x^{n} e^x} dx = \sum _{i=0} ^{n} (-1)^i\frac{n!}{(n-i)!}e^xx^{n-i} [/tex]
 
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  • #4
chanvincent is correct. when its the same integral with e^NEGATIVE x, its a nice definiton for the gamma function.
 

FAQ: I with finding the intergral of (x^n)(e^x)

What is the purpose of finding the integral of (x^n)(e^x)?

The purpose of finding the integral of (x^n)(e^x) is to determine the area under the curve of the function (x^n)(e^x) between two specific points, which can be useful in various applications such as calculating probabilities and solving differential equations.

How do I find the integral of (x^n)(e^x)?

To find the integral of (x^n)(e^x), you can use integration by parts or substitution. The specific method used will depend on the values of n and the complexity of the function. It is recommended to consult a calculus textbook or seek assistance from a tutor or teacher for step-by-step instructions.

What are the limits of integration for (x^n)(e^x)?

The limits of integration for (x^n)(e^x) will depend on the specific problem or application. In the case of indefinite integration, the limits will be from negative infinity to positive infinity. However, in definite integration, the limits will be determined by the given boundaries of the problem.

Can I use a calculator to find the integral of (x^n)(e^x)?

Some calculators have built-in integration functions, but they may not be able to handle complex functions like (x^n)(e^x). It is recommended to use a graphing calculator or a computer algebra system for more accurate results. However, it is important to understand the concepts and methods behind integration to verify the results.

What are the applications of (x^n)(e^x) and its integral?

The function (x^n)(e^x) and its integral have various applications in mathematics, physics, and engineering. For example, they can be used to model population growth, radioactive decay, and electric circuits. In economics, they can be used to calculate compound interest and the present value of an investment.

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