Integrating By Parts: Is 4 Times Best for \int {x^4 e^x dx}?

In summary, integrating by parts is a method used in calculus to find the integral of a product of two functions. It involves using the product rule of differentiation in reverse to rewrite the integral as a simpler one. When choosing which functions to differentiate and integrate, it is important to carefully consider which will lead to the simplest integral. Other numbers besides 4 can be used, but may result in a more complicated integral. This method can be used for integrals involving different combinations of functions, but the key is to simplify the integral as much as possible.
  • #1
danago
Gold Member
1,123
4
Hi. When evaluating an integral such as:

[tex]
\int {x^4 e^x dx}
[/tex]

Is integrating by parts 4 times the best method, or is there a more efficient way?

Thanks in advance,
Dan.
 
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  • #2
I think you are going to have to do parts 4 times.
 
  • #3
Well if the problem was a definite integral for example say:
[itex]\int_{0}^{1} x^4e^x dx [/itex]

you could easily make a reduction formula for easy calculations. But that is a different question altogether. As previously stated you would have to integrate by parts at 4 times
 
  • #4
Alright, thanks for the replies guys :smile:
 

FAQ: Integrating By Parts: Is 4 Times Best for \int {x^4 e^x dx}?

What is integrating by parts?

Integrating by parts is a method used in calculus to find the integral of a product of two functions. It is particularly useful when the integral involves a product of a polynomial and an exponential function.

How does integrating by parts work?

The method of integrating by parts involves using the product rule of differentiation in reverse. It allows us to rewrite the integral as a simpler one, which can be easily evaluated.

Why is 4 times the best choice for \int {x^4 e^x dx}?

When using the integrating by parts method, it is important to choose a function to differentiate and a function to integrate that will lead to a simpler integral. In this case, choosing the polynomial function x^4 as the one to differentiate and the exponential function e^x as the one to integrate leads to the simplest integral.

Can other numbers besides 4 be used for \int {x^4 e^x dx}?

Yes, other numbers can be used for \int {x^4 e^x dx}. However, choosing a number other than 4 may result in a more complicated integral that is more difficult to evaluate.

What if the integral involves a different combination of functions?

The integrating by parts method can still be used for integrals involving different combinations of functions. It is important to carefully choose which function to differentiate and which function to integrate in order to simplify the integral as much as possible.

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