What is the definision of lan transformans

  • Thread starter transgalactic
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In summary, if you have a function that is linear and has a constant derivative, you can use the "integration by substitution" method to integrate the function. If the derivative of the function is not a constant, you cannot use this method and must use the "quotient rule."
  • #1
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i can take every time all the function on the buttom
put it in a lan
an divide it in its derivative?

even if i got a fuction
1/(x+2)^2

its integral solves [(x+2)^-1]/-1

or i can open this equetion (a+b)^2=x^2+2*x*2+2^2
then put it all in a lan ln(x^2+2*x*2+2^2) and i will divide it by it derivative

2x+4

ln(x^2+2*x*2+2^2)/2x+4


what is my red sigh
how do i deside to what way to go?
 
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  • #2
transgalactic said:
i can take every time all the function on the buttom
put it in a lan
an divide it in its derivative?

even if i got a fuction
1/(x+2)^2

its integral solves [(x+2)^-1]/-1

or i can open this equetion (a+b)^2=x^2+2*x*2+2^2
then put it all in a lan ln(x^2+2*x*2+2^2) and i will divide it by it derivative

2x+4

ln(x^2+2*x*2+2^2)/2x+4


what is my red sigh
how do i deside to what way to go?
I hate to criticize anyone's English (since my (put the language of your choice here!) is awful) but I can't really understand what you are saying here. In particular, what do you mean by " lan"?

Here is my response to what I think you are asking:
Yes, it is true that the anti-derivative, or integral, of (x+2)-2 is
(x+2)-1 divided by -1: that's the power rule: -2+ 1= -1 (and it helps that the derivative of x+ 2 itself is 1).

If you had (3x+ 2)-2 to integrate, you can do much the same thing: (3x+2)-1 but now divided by the derivative of 3x+ 2 which is 3: (-1/3)(3x+2)-1. That's because if you differentiate (-1/3)(3x+2)-1, you would use both the "power rule" (with n-1= -1-1= -2) and the "chain rule" (the derivative of 3x+2 is 3) to multiply by -1 and 3 and change the power to -2: (-1)(3)(-1/3)(3x+2)-2, the original function.

However it is important there that the "inner" function, x+ 2 in the first case and 3x+2 in the second, is linear and its derivative is a constant.

The "chain rule in reverse" is really "integration by substitution": To integrate (3x+2)-2, more formally, let u= 3x+ 2. Then du= 3 dx or dx= (1/3)du and (3x+2)-2= u-2 so [itex]\int (3x+2)^{-2}dx= (1/3)\int u^{-2}du= (-1/3)u^{-1}+ C= (-1/3)(3x+2)^{-1}+C[/itex].

If that were (x2+ 2)-2 we cannot do that! If we try to let u= x2+ 2 then du= 2x dx and there is NO "2x" in the integral. While we could write (1/3)du= dx before, we cannot write (1/2x)du= dx because we can't have an "x" in the "u" integral.

In particular, to integrate (x2+ 4x+ 4)-1, which I think is your second example, we CANNOT just say "well since that is a -1 power, the anti-derivative is a logarithm and the we divide by the derivative of x2+ 4x+ 4= 2x" because that derivative is NOT a constant.
If you tried to make the substitution u= x2+ 4x+ 4, then du= (2x+ 4) dx and we cannot just divide by that!

Of course, you can always check an integral by differentiating;
What do you get if you differentiate ln(x2+ 4x+ 4)/(2x+ 4)?

You would have to use the quotient rule: the derivative of ln(x2+ 4x+ 4) time 2x+4 minus ln(x2+ 4x+ 4) times the derivative of 2x+ 4, all divided by (2x+4)2. That is
{(1/(x^2+ 4x+ 4)(2x+ 4)(2x+4)- ln(x2+ 4x+ 4)(2)}/(2x+4)2. That is not anything like your original (x2+ 4x+ 4)-1!

The rule "integrate f(x)n by integrating un and then dividing by the derivative of f works only if f is linear so its derivative is a constant. Otherwise see if its derivative is already in the integral so you can substitute.
 
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  • #3
so i which case we know that an integral equals to a ln fuction

and what are my options in the case of non lenear fraction like x^2+1
?

i know that one way is using trigonometric substitution

but what do i do i a case of e^x for example?
 
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What is the definition of lan transformation?

Lan transformation, also known as local area network transformation, refers to the process of upgrading or changing the technology, infrastructure, or design of a local area network (LAN) to improve its performance, security, or capabilities.

Why is lan transformation important?

Lan transformation is important for businesses and organizations because it allows for the modernization and optimization of their LAN, leading to improved efficiency, productivity, and communication among employees. It also helps to keep the network up-to-date with the latest technology and security measures.

What are the key elements of lan transformation?

The key elements of lan transformation include network assessment, planning, design, implementation, and testing. Network assessment involves analyzing the current LAN to identify areas for improvement. Planning and design involve creating a strategy and blueprint for the transformed LAN. Implementation is the actual process of making changes to the network, and testing ensures that the new LAN is functioning correctly.

What are some common challenges in lan transformation?

Some common challenges in lan transformation include budget constraints, compatibility issues with existing systems and devices, and downtime during implementation. It's important to carefully plan and communicate with all stakeholders to minimize these challenges.

How often should a lan transformation be conducted?

The frequency of lan transformation depends on the specific needs and goals of an organization. Some may need to undergo a transformation every few years to keep up with rapidly changing technology, while others may only need to do so every 5-10 years. It's important to regularly assess the network and make updates as needed to ensure optimal performance.

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