Integrating Perfect Cube in Denominator: 2/(x-2)^3 dx | Homework Solution

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In summary, Arun gave Cristo a formula for integrating a polynomial function of x. The formula is called the expansion formula and is applicable to negative powers as well.
  • #1
lost_math
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Homework Statement


\ 2/(x-2)^3 dx
Basically integrating a perfect cube in the denominator with a constant in the numerator


Homework Equations





The Attempt at a Solution


i thought it would be a form of ln(x), but then, that would mean having atleast some x terms in the numerator which are not there, so, how do i do this? Is there a known pre-fixed solution for these things? Like exp(something)?
 
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  • #2
Use [tex]\int x^ndx = \frac{x^{n+1}}{n+1} + c[/tex]
 
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  • #3
Try writing as 2(x-2)-3. Do you know how to integrate x-3?
 
  • #4
Thanks Arun and Cristo.
Cristo, I don't know how to integrate x^-3. (i think i might be hopeless, right?)
Arun, what is the expansion formula you just gave me called? Is there a name for solving by that method? It is applicable to negative powers as well?
 
  • #5
Arunbg's formula is the formula for integrating a polynomial function of x. Thus, to integrate x-3 you would use that formula.
 
  • #6
you want to integrate 2(x-2)-3

using this [tex]\int x^n = \frac{x^{n+1}}{n+1} + c[/tex]

the first thing you notice is that you add 1 to the power, then you divided by the new power
after that you multiply by the differntial of what is inside the brackets
 
  • #7
in your question the power is -3
 
  • #8
sara_87 said:
you want to integrate 2(x-2)-3

using this [tex]\int x^n = \frac{x^{n+1}}{n+1} + c[/tex]

the first thing you notice is that you add 1 to the power, then you divided by the new power
after that you multiply by the differntial of what is inside the brackets

The last bit should read: you divide by the derivative of the term inside the brackets.

(I know it's probably a typo, and it doesn't matter in this case; but it may confuse the OP in future if left uncorrected)
 
  • #9
yes definitely a typo :blushing:
you always divid by the differential of the inside of the brackets when integrating!
 

1. What is a perfect cube?

A perfect cube is a number that can be expressed as the product of three equal factors. For example, 8 is a perfect cube because it can be expressed as 2 x 2 x 2.

2. Why is it important to integrate a perfect cube in the denominator?

Integrating a perfect cube in the denominator allows us to solve for a more complex integral by using techniques such as u-substitution. It also helps to simplify the integral and make it more manageable.

3. How do you integrate a perfect cube in the denominator?

To integrate a perfect cube in the denominator, we can use the substitution u = x-2. This will allow us to rewrite the integral in terms of u and then use the power rule to solve it.

4. Can you give an example of integrating a perfect cube in the denominator?

Yes, for example, if we have the integral 2/(x-2)^3 dx, we can use the substitution u = x-2. This will give us the integral 2/u^3 du. We can then solve this using the power rule, which results in the final answer of -1/u^2 + C. Substituting back in u = x-2, we get the final answer of -1/(x-2)^2 + C.

5. What is the significance of the dx in the integral 2/(x-2)^3 dx?

The dx in the integral represents the variable with respect to which we are integrating. In this case, it represents the variable x. This allows us to find the area under the curve of the function 2/(x-2)^3 with respect to x.

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