Antiderivative of 1/(1+e^x) - kinematics question

In summary, the conversation discusses finding the antiderivative of a given function in terms of another variable. The conversation includes various approaches and suggestions for solving the problem, ultimately resulting in the solution of x in terms of t.
  • #1
meee
87
0
Antiderivative of this? - kinematics

a kinematics question

v = 1 + e^x

find x in terms of t given that x = 0 when t = 0

what i did:

dx/dt = 1 + e^x
dt/dx = 1/ (1+ e^x)

so t = the antiderivative of 1/(1+ e^x)

i tried and i kinda did this...

loge(1 + e^x)/e^x + cbut it doesn't seem right
 
Last edited:
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  • #2
Notice that if you had an e^x in the numerator of your integral, it will be easy.

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

Can you take it from here?
 
  • #3
yeah that helped thanks... i got to t= x + loge(1+e^x)
but how do i get x in terms of t?
 
  • #4
HINT:

[tex]\int \frac{f'(x)}{f(x)} dx = \ln\left| f(x) \right| + C[/tex]
 
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  • #5
That's wrong Hoot, I think you missed a f'(x).

To get x in terms of t is painful. Write x as log(e^x), you'll get a quadratic in e^x, which you can solve.
 
  • #6
siddharth said:
That's wrong Hoot, I think you missed a f'(x).
Indeed I have, don't know what I was thinking to be honest. Thanks again for you contribution to the pH tutorial.
 
  • #7
thanks gusys
 
  • #8
No problem, Hoot.

meee, I think I found a couple of errors in your work. You missed a minus sign, and your constant of integration. It should be

t= x - loge(1+e^x) + C

Put the initial conditions (ie, x=0 when t=0) to find C.

To get x in terms of t, write x as log(e^x) and play around for a while.
 
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  • #9
oh right... thanks!
 

1. What is an antiderivative?

An antiderivative is the inverse operation of a derivative. It is a mathematical function that, when differentiated, returns the original function.

2. How do you find the antiderivative of a function?

To find the antiderivative of a function, you can use a variety of techniques such as the power rule, integration by parts, or substitution. It is important to note that the antiderivative of a function is not unique, as it can have multiple solutions.

3. What is the difference between an antiderivative and an indefinite integral?

An antiderivative is a specific function that, when differentiated, returns the original function. An indefinite integral, on the other hand, is a family of functions that differ from each other by a constant. In other words, an indefinite integral is a set of all possible antiderivatives of a given function.

4. Can all functions have an antiderivative?

No, not all functions have an antiderivative. For a function to have an antiderivative, it must be continuous on its domain. In addition, some functions, such as trigonometric and exponential functions, have specific rules for finding their antiderivatives.

5. How can antiderivatives be used in real-life applications?

Antiderivatives have numerous applications in different fields of science and engineering. For example, in physics, antiderivatives are used to calculate displacement and velocity from acceleration, while in economics, they are used to model and predict changes in supply and demand. They are also used in various engineering disciplines to solve problems involving rates of change.

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