Integrating the Inverse Exponential-Square Root Function

In summary, users are discussing a problem involving finding the anti-derivative of a given expression. They suggest using a substitution, and provide hints on how to proceed. The final answer is debated, with one user suggesting cos^-1(e^-x) + C and another suggesting -sin^-1(e^-x) + C.
  • #1
physnoob
15
0

Homework Statement


[tex]\int dx/(e^{x}\sqrt{1-e^{-2x}}) [/tex]

Homework Equations





The Attempt at a Solution


I have absolutely no idea of how to start the problem, any help is greatly appreciated!
thanks!
 
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  • #2
Try a substitution of maybe u=e-x. Then you should get it into a form where the anti-derivative should be easily found.
 
  • #3
Also, note that 1/ex = e-x.
 
  • #4
hmm, can i get a little more hint? if i do a u sub. of u = e[tex]^{-x}[/tex], how do i get rid of
e[tex]^{-2x}[/tex]? I have tried do a u sub. of u = [tex]\sqrt{1-e^{-2x}}[/tex], but i ended up getting the [tex]\int du/e^{-x}[/tex], which i don't know how to proceed after this. What am i doing wrong?
 
  • #5
e-2x=(e-x)2, so in terms of u it is?
 
  • #7
o boy, that was embarrassing lol
just want to make sure, is the answer cos[tex]^{-1}[/tex](e[tex]^{-x}[/tex]) + C?
 
  • #8
That should be correct.

I think -sin-1(e-x)+C should work as well.
 

1. What is the inverse exponential-square root function?

The inverse exponential-square root function is a mathematical function that is used to find the input value (or "argument") that will result in a given output value. It is the inverse of the exponential-square root function f(x) = e^(√x), which takes the square root of the input value and then raises it to the power of the base of the natural logarithm, e.

2. How is the inverse exponential-square root function integrated?

To integrate the inverse exponential-square root function, we use the substitution method. We substitute u = √x and du = 1/(2√x)dx to transform the function into ∫(1/u)e^u du. This can then be integrated using the power rule for integration, resulting in the final solution of e^u + C. Finally, we substitute back in the original variable x to get the final integral of e^(√x) + C.

3. What is the domain and range of the inverse exponential-square root function?

The domain of the inverse exponential-square root function is all positive real numbers, since the square root of a negative number is not defined. The range of the function is also all positive real numbers, since the exponential function grows exponentially as the input value increases.

4. Why is the inverse exponential-square root function important?

The inverse exponential-square root function is important in many areas of mathematics and science, as it allows us to find the input value that results in a given output value. This is useful in solving equations, finding maximum and minimum values, and in many real-life applications such as population growth and radioactive decay.

5. Can the inverse exponential-square root function be graphed?

Yes, the inverse exponential-square root function can be graphed. Its graph is a curve that increases rapidly at first and then levels off as the input value increases. It approaches but never reaches the x-axis, as the output value will never be equal to 0. The graph also has a vertical asymptote at x = 0, since the function is not defined for negative values of x.

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