Integral - an alternative to expanding the denominator?

In summary, the conversation discusses an integral that can be solved by expanding the denominator, but the speaker wonders if there is another way to solve it. The conversation then goes on to mention the possibility of using integration by substitution or factoring and using partial fractions. However, it is acknowledged that these methods may not necessarily be considered "easy."
  • #1
LikeMath
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Integral -- an alternative to expanding the denominator?

The following integral can be easily solved by expanding the denominator but I am wondering if there is a another way to solve it

[itex]\int\frac{1}{(x+1)^7-x^7-1}dx[/itex]
 
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  • #2


You could try doing integration by substitution, and then repeating it several (probably 7) times. But I'm sure you would either end up with something really ugly or encounter a problem pretty early on that stops you.
 
  • #3


LikeMath said:
The following integral can be easily solved by expanding the denominator but I am wondering if there is a another way to solve it

[itex]\int\frac{1}{(x+1)^7-x^7-1}dx[/itex]



Would you be so kind as to show us how this integral is "easily" solved expanding whatever? I think this is a rather

horrible integral, expanding or not, and more than finding a way to make it more or less normal I'd love to see what's your way to solve it.

Thanx

DonAntonio
 
  • #4


DonAntonio said:
Would you be so kind as to show us how this integral is "easily" solved expanding whatever? I think this is a rather

horrible integral, expanding or not, and more than finding a way to make it more or less normal I'd love to see what's your way to solve it.

Thanx

DonAntonio

By the binomial theorem we get
[itex](x+1)^7-x^7-1=7(x^6+3x^5+5x^4+3x^2+x)[/itex]
now if we factorize this term we also get
[itex]x(x+1)(x^2+x+1)^2[/itex]
then partial fraction completes the solution.
 
  • #5


LikeMath said:
By the binomial theorem we get
[itex](x+1)^7-x^7-1=7(x^6+3x^5+5x^4+3x^2+x)[/itex]
now if we factorize this term we also get
[itex]x(x+1)(x^2+x+1)^2[/itex]
then partial fraction completes the solution.


Well, yes...but for this you must first (1) know how to factorize the polynomial (a quintic, since zero is obvious), perhaps by "guessing

that -1 is a root, and (2) you must still make the partial fractions stuff, which seems far from being that easy, as [tex]\frac{1}{x(x+1)(x^2+x+1)^2}=\frac{A}{x}+\frac{B}{x+1}+\frac{Cx+D}{x^2+x+1}+\frac{Ex+F}{(x^2+x+1)^2}[/tex]
A matter of taste, I guess...perhaps because I'm a theoretical mathematician I wouldn't dare call the above "easy", or perhaps I would

but I'd add immediately "annoying and long" after that.

DonAntonio
 
  • #6


Yes you are right, "easy" was not appropriate.
 
  • #7


Hi !
Who said "it's not easy" ?
It takes less time to compute it that to type it. :redface:
 

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  • #8


JJacquelin said:
Hi !
Who said "it's not easy" ?
It takes less time to compute it that to type it. :redface:



Really...?! Common, it's nice to show off but not with this petty things.

DonAntonio
 

Related to Integral - an alternative to expanding the denominator?

1. What is the concept of "Integral - an alternative to expanding the denominator" in mathematics?

The concept of "Integral - an alternative to expanding the denominator" is a mathematical technique used to solve integrals involving fractions with polynomials in the denominator. It involves rewriting the fraction as a sum of simpler fractions, each with a different denominator, in order to make it easier to integrate.

2. How is "Integral - an alternative to expanding the denominator" different from traditional methods of solving integrals?

"Integral - an alternative to expanding the denominator" is different from traditional methods of solving integrals because it allows for the integration of more complex fractions by breaking them down into simpler fractions. This method is particularly useful when the traditional methods of integration, such as substitution or integration by parts, are not applicable.

3. What are the advantages of using "Integral - an alternative to expanding the denominator" in solving integrals?

One of the main advantages of using "Integral - an alternative to expanding the denominator" is that it allows for the integration of more complex fractions, which may not be possible using traditional methods. It also simplifies the integration process and can lead to more concise and elegant solutions.

4. Are there any limitations or drawbacks to using "Integral - an alternative to expanding the denominator"?

One limitation of using "Integral - an alternative to expanding the denominator" is that it may not always be applicable or may not lead to a simpler integration process. It also requires a good understanding of algebra and factoring to be used effectively.

5. How can "Integral - an alternative to expanding the denominator" be applied in real-world situations?

"Integral - an alternative to expanding the denominator" can be applied in various fields such as physics, engineering, and economics, where integrals involving fractions with polynomials in the denominator often arise. It can help in solving problems related to optimization, finding areas and volumes, and modeling real-life situations.

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