# Integrating the Inverse Exponential-Square Root Function

• physnoob
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.
physnoob

## Homework Statement

$$\int dx/(e^{x}\sqrt{1-e^{-2x}})$$

## The Attempt at a Solution

I have absolutely no idea of how to start the problem, any help is greatly appreciated!
thanks!

Try a substitution of maybe u=e-x. Then you should get it into a form where the anti-derivative should be easily found.

Also, note that 1/ex = e-x.

hmm, can i get a little more hint? if i do a u sub. of u = e$$^{-x}$$, how do i get rid of
e$$^{-2x}$$? I have tried do a u sub. of u = $$\sqrt{1-e^{-2x}}$$, but i ended up getting the $$\int du/e^{-x}$$, which i don't know how to proceed after this. What am i doing wrong?

e-2x=(e-x)2, so in terms of u it is?

Doesn't e-x*e-x = e-2x?

o boy, that was embarrassing lol
just want to make sure, is the answer cos$$^{-1}$$(e$$^{-x}$$) + C?

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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