Solve Integral Error: \int\frac{1+e^x}{1-e^x}dx

In summary, the conversation discusses solving the integral \int \frac{1+e^x}{1-e^x}dx using substitution. The first integral is solved using the substitution w = e^x, while the second integral is solved using u = 1-e^x. The final result is x - 2\ln|1-e^x| + C. The conversation also mentions the use of natural logs and absolute values in solving the integral.
  • #1
G01
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[tex] \int \frac{1+e^x}{1-e^x}dx = [/tex]

[tex] \int \frac{dx}{1-e^x} + \int \frac{e^x}{1-e^x}dx [/tex]

The second integral can be done by the substitution
[tex] u = 1-e^x [/tex]
[tex] du = -e^x dx[/tex]

So the second integral becomes:
[tex]\int \frac{du}{u} = \ln|u|+C[/tex]
In the first integral, you can use the substituion:
[tex]w = e^x[/tex]
[tex]dw = e^x dx[/tex]

So the first integral becomes:
[tex] \int \frac{dw}{w(1-w)} [/tex]
This can be done by parts and you get:
[tex] \int \frac{dw}{w} + \int \frac{dw}{1-w} [/tex]

These are also both natural logs so you end up with:

[tex] \ln|e^x| - 2\ln|1-e^x| + C [/tex]

[tex] x - 2\ln|1-e^x| +C [/tex]

I had http://integrals.wolfram.com/index.jsp compute this for me and it got:

[tex] x - 2\ln(e^x - 1) +C [/tex]

I think this may have something to do with the absolute value, but I always make stupid mistakes with signs so I thought I'd check...Thanks for the time I probably just wasted.
 
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  • #2
For some reason it won't let me edit my post... on the fifth line that integral should be
[tex]-\int\frac{du}{u} [/tex]
 
  • #3
sure, |a-b|=|b-a|
 
  • #4
ok thanks alot, stupid question
 

1. What is the purpose of solving an integral error?

Solving an integral error allows us to accurately calculate the area under a curve, which is important in many scientific and mathematical applications.

2. How do you approach solving an integral error?

The first step is to rewrite the integral in a more manageable form, such as using substitution or partial fractions. Then, we can use different techniques such as integration by parts or trigonometric substitution to solve the integral.

3. What is the specific error in the given integral, and how can it be solved?

The specific error in this integral is the presence of a singularity at x = ln(1), where the denominator becomes 0. This can be solved by using a substitution such as u = 1-e^x to remove the singularity and allow for a proper integration.

4. Are there any common mistakes to watch out for when solving an integral error?

Yes, some common mistakes include not properly identifying and handling singularities, making algebraic errors during integration, and forgetting to add the constant of integration at the end.

5. Can technology be used to solve an integral error?

Yes, there are many mathematical software and calculators that can solve integrals, including integral errors. However, it is still important to understand the concepts and steps involved in solving the integral to ensure accuracy and avoid mistakes.

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