Integrating a Polynomial with Fractional and Negative Indices

In summary, the problem with the question is that the x was eliminated from the last term of the equation.
  • #1
_Mayday_
808
0
I have a bit of a problem with this question, I will do my best to offer an answer. I think the problem is not with the differentiation but with my indices. :smile:

Here is the initial formula.

[tex]\int (6x + 2 + x^{-\frac{1}{2}}) dx [/tex]

Here is my attempt :blushing:

[tex]\frac{6x^2}{2} + 2x + \frac{x\frac{1}{2}}{\frac{1}{2}}[/tex]

[tex]3x^2 + 2x + \frac{\sqrt\frac{1}{2}}{\frac{1}{2}} + c[/tex]


Where have I gone wrong? I'll bet it's the whole indices thing!

Thanks for any help :smile:
 
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  • #2
Do you mean:

[tex]\frac{\sqrt{x}}{\frac{1}{2}} = 2\sqrt{x}[/tex]

I don't see how the x disappeared in the last term of your solution.
 
  • #3
_Mayday_ said:
Here is my attempt :blushing:

[tex]\frac{6x^2}{2} + 2x + \frac{x\frac{1}{2}}{\frac{1}{2}}[/tex]

[tex]3x^2 + 2x + \frac{\sqrt\frac{1}{2}}{\frac{1}{2}} + c[/tex]

The problem is in the last term: that should be

[tex]\frac{x^{\frac{1}{2}}}{\frac{1}{2}}[/tex] ,

which is to say, x to the one-half power divided by one-half. X to the one-half power is the square root of x, but in any case, what you did in the next line was to drop the "x". You want to say

[tex]\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 {x^{\frac{1}{2}}[/tex] .
 
  • #4
dynamicsolo said:
[tex]\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 {x^{\frac{1}{2}}[/tex] .
Ah ok, why does it do that then? I don't understand that could you take me through it please.

Thanks to both of you so far :smile:
 
  • #5
Do you mean, why is dividing by 1/2 equal to multiplying by 2?
 
  • #6
Well, how does this work?

[tex]\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 {x^{\frac{1}{2}}[/tex] .
 
  • #7
_Mayday_ said:
Well, how does this work?

[tex]\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 {x^{\frac{1}{2}}[/tex] .

You want to "eliminate" the 1/2 in the denominator of this quotient, which means making it 1, so that you end up with the numerator divided by 1, which is just the numerator. You can multiply the numerator and denominator by 2, which is like multiplying your quotient by 2/2 = 1, which means the value of the fraction is unchanged. You will end up with

[tex]\frac{2x^{\frac{1}{2}}}{1} = 2 {x^{\frac{1}{2}} = 2\sqrt{x}[/tex]
 
Last edited:
  • #8
dynamicsolo said:
You want to "eliminate" the 1/2 in the denominator of this quotient, which means making it 1, so that you end up with the numerator divided by 1, which is just the numerator. You can multiply the numerator and denominator by 2, which is like multiplying your quotient by 2/2 = 1, which means the value of the fraction is unchanged. You will end up with

[tex]\frac{2x^{\frac{1}{2}}}{1} = 2 {x^{\frac{1}{2}} = 2\sqrt{x}[/tex]

Brilliant! Thanks for your time you two! Funny how trivial these things seem in hind sight! :shy:

Thanks Again!
 
  • #9
So then my final equation will be:

[tex]3x^2 + 2x + 2x^{\frac{1}{2}} + c[/tex]
 
  • #10
_Mayday_ said:
So then my final equation will be:

[tex]3x^2 + 2x + 2x^{\frac{1}{2}} + c[/tex]

If you want to check it, differentiate it term-by-term: you'll get your original integrand back again... (It's correct!)
 

What is an integration index?

An integration index is a numerical measure used to assess the level of integration within a system or between different systems. It is often used in fields such as ecology, economics, and sociology to study the connectivity and interactions between different elements.

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How are integration indices calculated?

The specific calculation method for integration indices varies depending on the index being used. However, in general, integration indices are calculated by taking into account various factors such as the number and relative abundance of different elements, their level of interaction, and their diversity or similarity.

What are the limitations of integration indices?

Integration indices are useful tools for measuring the level of integration within a system, but they also have some limitations. For example, they may not accurately reflect changes in the underlying systems over time, and they may not account for all relevant factors that contribute to integration.

How can integration indices be applied in other fields?

While integration indices are commonly used in ecology, they can also be applied in other fields such as economics, sociology, and urban planning. In these fields, integration indices can be used to measure the level of interconnectedness between different elements, such as economic systems, social networks, or urban infrastructure.

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