Integral with e^x: Solving for ln(e^2x + 1)

In summary, the conversation is about a problem with the integral of (2e^x)/(e^x+e^-x)dx and the desired result of ln(e^2x + 1). The solution involves simplifying the problem and using u-substitution, with the derivative of e^2x being a key component. The mistake was likely made in this step. The solution involves multiplying the numerator and denominator by e^x and using the substitution t = e^2x + 1 to arrive at the desired result of ln(e^2x + 1).
  • #1
Michael Gulley
1
0
The problem I have is

∫(2e^x)/(e^x+e^-x)dx

I cannot seem to get to the correct result, ln(e^2x + 1). I always have (e^2x)ln(e^2x + 1). What do I need to do, to get rid of the e^2x.
Any help would be greatly appreciated.
 
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  • #2
Do the problem correctly. How's that for a nice vague reply? Unfortunately, that's about all anyone can say without seeing your work. How are we supposed to see where you're going wrong if you don't show your work?
 
  • #3
Step 1. Try to simplify the problem (i.e., take any constants out of the integral).

Step 2. You may want to try u-substitution.
 
  • #4
Hint: What's the derivative of e^2x?

I think this is probably where the mistake was made.
 
  • #5
Its simple. Multiply your numerator and denominator by e^x. This will get you (2e^2x)/(e^2x + 1)dx. Now suppose your whole denominator as another variable,say, t. Calculate dt which will be equal to 2e^2xdx.
So now your integral is in the form of (1/t)dt. Integration of this will give you ln(t). Substitute back the value of 't' and there you have it.
 

FAQ: Integral with e^x: Solving for ln(e^2x + 1)

What is the purpose of solving for ln(e^2x + 1) when using integrals with e^x?

The purpose of solving for ln(e^2x + 1) is to find the antiderivative of the function e^2x + 1. This allows for the calculation of the definite integral, which can provide useful information about the original function.

What is the process for solving for ln(e^2x + 1) using integrals with e^x?

The process for solving for ln(e^2x + 1) involves using the substitution method. The variable u is substituted for the expression inside the natural logarithm, and the derivative of u is used to replace the differential dx. This will then simplify the integral and allow for easier integration.

Can ln(e^2x + 1) be solved without using the substitution method?

Yes, ln(e^2x + 1) can also be solved using integration by parts. This method involves splitting the integral into two parts, one of which becomes the new u and the other becomes the new dv. This method can be used as an alternative to the substitution method.

What is the importance of using e^x in the integral with ln(e^2x + 1)?

The use of e^x in the integral with ln(e^2x + 1) is important because it is the base of the natural logarithm. This allows for the simplification of the integral and makes the integration process easier.

Are there any real-world applications of solving for ln(e^2x + 1) using integrals with e^x?

Yes, there are many real-world applications of solving for ln(e^2x + 1) using integrals with e^x. Some examples include calculating the growth rate of bacteria, determining the rate of change in population growth, and predicting the rate of decay of radioactive substances.

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