- #1

hadi amiri 4

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Can anyone help me with this [tex]\int \frac{dx}{x^7-x}[/tex]?

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- Thread starter hadi amiri 4
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In summary, the conversation discusses the simplification of the expression \int \frac{dx}{x^7-x} by re-writing the denominator, using polynomial division, and possibly factoring and using partial fractions decomposition. There is also a suggestion for a simpler method, but it is deemed to be too trivial. The final result is \int \frac{dx}{x^7-x} = -ln(x)+(1/6)*ln(x^6-1).

- #1

hadi amiri 4

- 98

- 1

Can anyone help me with this [tex]\int \frac{dx}{x^7-x}[/tex]?

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- #2

arildno

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1. Re-write the denominator as x*(x^6-1)=x*(x^3-1)*(x^3+1)

2. Use polynomial division to reduce the third-degree polynomials:

[tex](x^{3}\pm{1}):(x\pm{1})=x^{2}\mp{x}+1[/tex]

3. See if these can be factorized any further, then use partial fractions decomposition.

- #3

hadi amiri 4

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but it takes an hour to do it

is there any simpler method for this?

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HallsofIvy

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I guess you had better stick to trivial problems!

- #5

saeed69

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1/(x^7-x) = 1/(x*(x^6-1))= -1/x+x^5/(x^6-1)

int(-1/x+x^5/(x^6-1), x)=

int(-1/x, x)=-ln(x)

int(x^5/(x^6-1), x)=(1/6)*ln(x^6-1)

int(1/(x^7-x), x)=-ln(x)+(1/6)*ln(x^6-1)

- #6

arildno

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I didn't see that one..1/(x^7-x) = 1/(x*(x^6-1))= -1/x+x^5/(x^6-1)

Very good, saeed69!

When we talk about "re-writing the denominator," we are referring to changing the expression in the denominator of a fraction to a different form that may be easier to work with. In this case, we are looking at an expression with a denominator of `x*(x^6-1)`

and we want to re-write it as `x*(x^3-1)*(x^3+1)`

.

Re-writing the denominator can help us simplify or manipulate a fraction in order to solve a problem or answer a question. In this case, re-writing the denominator as `x*(x^3-1)*(x^3+1)`

may help us to more easily factor or cancel out terms in the fraction.

In general, there is no set formula for re-writing a denominator. It often involves analyzing the expression and looking for patterns or relationships between the terms. In this case, we can see that the expression `x^6-1`

can be factored into `(x^3-1)*(x^3+1)`

, which allows us to re-write the denominator as `x*(x^3-1)*(x^3+1)`

.

To re-write the denominator in this case, we first need to identify any common factors between the terms. In this example, we can see that both `x^6`

and `1`

can be factored into `x^3`

and `1`

, respectively. Then, we can use the distributive property to factor out the common factor of `x^3`

, resulting in the expression `x*(x^3-1)*(x^3+1)`

.

Re-writing the denominator does not change the overall value of the expression, but it may help us to simplify or manipulate it in order to solve a problem or answer a question. In this case, re-writing the denominator as `x*(x^3-1)*(x^3+1)`

allows us to factor or cancel out terms in the fraction, which may make it easier to work with.

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