Integrate (e^x +1)/(e^x -1): Tips & Hints

In summary, the conversation is about integrating (e^x + 1)/(e^x - 1) using substitution. One suggested method is to break the integral into two pieces and use different substitutions on each piece. Another suggestion is to use partial fractions or to let u = e^x - 1. Different methods may result in different forms of answers.
  • #1
Swatch
89
0
Hi. I am trying to integrate (e^x +1)/(e^x -1)
I have looked at this for almost an hour and don´t know how to start with it. I want to use substitution but I have to rewrite this in some way. Could anyone please give me a little hint?
 
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  • #2
Swatch, try breaking the integral into two pieces and using a different substitution on each piece. One of the pieces will immediately look easy, and the other will be easy after one line of algebra.
 
  • #3
Perhaps I'm missing something but a kind of obvious substitution would be
u= ex. Then (ex+1)/(ex-1) becomes (u-1)/(u+ 1). Of course, du= exdx so dx= (1/u)du. The integral
[tex]\int \frac{e^x+1}{e^x-1}dx[/tex] becomes
[tex]\int \frac{u+1}{u(u-1)}du[/tex]
which can be done with partial fractions.
If you don't like partial fractions, let u= ex-1. Then du= exdx so dx= (1/u) du again but now ex+ 1= u+ 2. The integral becomes
[tex]\int\frac{u+2}{u^2}du= \int \left(u^{-1}+ 2u^{-2}\right)du[/tex].
 
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  • #4
I integrated with u=e^x and used partial fraction
to get
-ln(e^x) + 2*ln(e^x -1) +C

I differentiated this back to the beginning, so I should be right. But I got a different looking answer in my textbook.
 
  • #5
If the derivative of your solution gives the integrand again, you should be OK.
It's possible to find different forms of answers when integrating, that very much depends on the method used.
 
  • #6
thank you for your help
 
  • #7
mathmatica says it is

-2x+xln(x)+Li(x)
 

FAQ: Integrate (e^x +1)/(e^x -1): Tips & Hints

What is the purpose of integrating (e^x + 1)/(e^x - 1)?

The purpose of integrating (e^x + 1)/(e^x - 1) is to find the antiderivative or the original function that, when differentiated, would result in the given expression. This is useful in solving various mathematical problems and in the fields of physics and engineering.

What are the steps involved in integrating (e^x + 1)/(e^x - 1)?

The steps involved in integrating (e^x + 1)/(e^x - 1) are as follows:

  1. Use the substitution method to rewrite the expression as (e^x - 1 + 2)/(e^x - 1), where 2 is a constant.
  2. Expand the expression to get e^x - 1 + 2/(e^x - 1).
  3. Integrate e^x - 1 using the power rule, which gives e^x - x + C, where C is the constant of integration.
  4. Integrate 2/(e^x - 1) using the natural logarithm rule, which gives 2ln|e^x - 1| + C.
  5. Combine the two results to get the final answer, e^x - x + 2ln|e^x - 1| + C.

What are some tips for integrating (e^x + 1)/(e^x - 1)?

Some tips for integrating (e^x + 1)/(e^x - 1) are:

  • Make use of algebraic manipulation, such as the substitution method, to simplify the expression before integrating.
  • Be familiar with the power rule and the natural logarithm rule, as these are the most commonly used integration rules for this type of expression.
  • Double-check your work by differentiating the final result to ensure that it matches the original expression.
  • Practice and familiarize yourself with different types of integrals to improve your skills.

What are some common mistakes to avoid when integrating (e^x + 1)/(e^x - 1)?

Some common mistakes to avoid when integrating (e^x + 1)/(e^x - 1) are:

  • Forgetting to add the constant of integration, which is necessary when integrating indefinite integrals.
  • Making errors in algebraic simplification, which can lead to incorrect results.
  • Not being familiar with the basic integration rules, leading to incorrect application of the rules.
  • Forgetting to substitute back in the original variable after using the substitution method.

How can I check if my answer for integrating (e^x + 1)/(e^x - 1) is correct?

You can check if your answer for integrating (e^x + 1)/(e^x - 1) is correct by differentiating the result and seeing if it matches the original expression. You can also use online integration calculators or ask for peer or instructor feedback to verify your work.

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