Area Under ArcTan[x] - Calculate F(x)

  • Thread starter PrudensOptimus
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In summary, the arc tan function can be approximated by using a MacLauren series. The convergence of the series can be complicated to calculate, but it can be done.
  • #1
PrudensOptimus
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Any errors? Please pick out and explain, thanks.

[tex]\int{}tan^{-1}(x)dx = F(x)[/tex]
[tex]F'(x) = tan^{-1}(x)[/tex]
[tex]\frac{dy}{dx} = tan^{-1}(x)[/tex]
[tex]dy = tan^{-1}(x) dx[/tex]
[tex]tan^{-1}(\frac{dy}{dx}) = tan(x) [/tex]
[tex]\frac{F'(x)}{1+F^{2}(x)} = sec^{2}(x)[/tex]
[tex]F'(x) = sec^{2}(x)[1 + F^{2}(x)][/tex]
[tex]F(x) = tan(x) + \int{}\frac{sin(x)}{cos^{3}(x)}dx[/tex]
[tex]F(x) = tan(x) + \frac{1}{2cos^{2}(x)} + C[/tex]
 
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  • #2
arctg(dy/dx)=tgx

can you show me arctg(dy/dx)=tgx?



your question is very interesting!
 
  • #3


Originally posted by kallazans
can you show me arctg(dy/dx)=tgx?



your question is very interesting!


I think I already typed out all my steps...
 
  • #4
This step is wrong:
[tex]dy = tan^{-1}(x) dx[/tex]
[tex]tan^{-1}(\frac{dy}{dx}) = tan(x) [/tex]

[tex]\frac{dy}{dx}= tan^{-1}(x)[/tex] so
[tex]tan(\frac{dy}{dx})= x[/tex]
 
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  • #5
Originally posted by HallsofIvy
This step is wrong:
[tex]dy = tan^{-1}(x) dx[/tex]
[tex]tan^{-1}(\frac{dy}{dx}) = tan(x) [/tex]

[tex]\frac{dy}{dx}= tan^{-1}(x)[/tex] so
[tex]tan(\frac{dy}{dx})= x[/tex]

ahh, what was i thinking:\
 
  • #6
Any other way to find ArcTan[x] area? No Parts please.
 
  • #7
Hi,

Why not using integration by parts? It's easy to do it that way.

Sam
 
  • #8
Originally posted by sam2
Hi,

Why not using integration by parts? It's easy to do it that way.

Sam

Well, sometimes the other way might define a new method of solving harder problems.
 
  • #9
I guess you could expand it in a series and integrate each term, then pick up the pieces again. If the series wouldn't turn out to be infinite as in this case all will be swell.
 
  • #10
Originally posted by Sonty
I guess you could expand it in a series and integrate each term, then pick up the pieces again. If the series wouldn't turn out to be infinite as in this case all will be swell.


Isn't Tan(x) and Tan^-1(X) a MacLauren series?
 
  • #11
you can expand around 0, of course, or around any other point. The annoying thing is that in the end you have to make all those convergence calculations. you can even go into a Fourier expansion so you won't be integrating polynomials, but cos and sin. whatever. you can always find harder ways to solve simple problems.
 

1. What is the purpose of calculating the area under ArcTan[x]?

The purpose of calculating the area under ArcTan[x] is to find the total area under the curve of the function ArcTan[x]. This can be used to determine the average value of the function or to solve real-world problems in fields such as physics, engineering, and economics.

2. How do you calculate the area under ArcTan[x]?

To calculate the area under ArcTan[x], you can use the formula A = ∫tan^-1(x)dx, where A is the area and tan^-1(x) is the inverse tangent function. This can be solved using integration techniques such as substitution or integration by parts.

3. What is the relationship between the area under ArcTan[x] and the function itself?

The area under ArcTan[x] is directly related to the function itself. The area represents the total value of the function over a given interval, with the x-axis as the lower boundary and the curve of the function as the upper boundary. As the function increases or decreases, the area under the curve will also change accordingly.

4. Can the area under ArcTan[x] be negative?

Yes, the area under ArcTan[x] can be negative. This occurs when the function produces negative values over a certain interval, resulting in a negative area. However, the absolute value of the area can still be used to determine the total value of the function over that interval.

5. How is the area under ArcTan[x] used in real-life applications?

The area under ArcTan[x] has many real-life applications, such as in physics to calculate the work done by a force, in engineering to determine the stress-strain relationship in materials, and in economics to calculate the total cost or revenue of a business. It can also be used in statistics to calculate the average value of a dataset or to determine the probability of an event.

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