Integrating e^x/(e^{2x} + 1): Long Division?

In summary, the purpose of long division in integrating e^x/(e^{2x} + 1) is to simplify the expression and make it easier to integrate. To perform long division, divide the numerator (e^x) by the denominator (e^{2x} + 1) using the same steps as traditional long division, resulting in a quotient and remainder. The remainder can then be written as a fraction with the original denominator and integrated separately. While it is possible to integrate the original expression without long division, it is more complicated and time-consuming. However, long division may not be necessary if the degree of the numerator is less than the degree of the denominator.
  • #1
RadiationX
256
0
i'm having trouble rewriting this integral:[tex]\int\frac{e^x}{e^{2x} + 1}[/tex] so that it will be in the arctan formula: should i use long divison here? if it were not for the [tex]e^x[/tex] in the numerator i'd be fine.
 
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  • #2
[tex] u = e^{x}, du = e^{x}dx[/tex]

[tex] e^{2x} = (e^{x})^2 [/tex]
 
  • #3
you never get away from u -substitutions do ! thanks
 
  • #4
Why woudl you watn to get away from them? They save your ass a lot :)
Learn to love em.
 
  • #5
yeah your right
 

Related to Integrating e^x/(e^{2x} + 1): Long Division?

1. What is the purpose of long division in integrating e^x/(e^{2x} + 1)?

The purpose of long division in this integration is to simplify the expression and make it easier to integrate.

2. How do I perform long division for this integration?

To perform long division, divide the numerator (e^x) by the denominator (e^{2x} + 1) using the same steps as traditional long division. This will result in a quotient and a remainder.

3. What do I do with the remainder after performing long division?

The remainder can be written as a fraction with the original denominator (e^{2x} + 1). This fraction can then be integrated separately.

4. Can I skip the long division and still integrate the expression?

It is possible to integrate the original expression without performing long division, but it will be more complicated and time-consuming. Long division simplifies the integration process.

5. Are there any special cases where long division may not be necessary for this integration?

Yes, if the degree of the numerator (e^x) is less than the degree of the denominator (e^{2x} + 1), then long division is not necessary and the integration can be done directly.

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