- #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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Re-write the denominator as x*(x^6-1)=x*(x^3-1)*(x^3+1)
- 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).
- #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?
- #4
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..
Very good, saeed69!
FAQ: Re-write the denominator as x*(x^6-1)=x*(x^3-1)*(x^3+1)
1. What does it mean to "re-write the denominator"?
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).
2. Why is it necessary to re-write the denominator?
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.
3. How do you know what form to re-write the denominator in?
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).
4. Can you explain the steps to re-write the denominator?
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).
5. How does re-writing the denominator affect the overall expression?
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.