Re: Integral involving square root of e^x

In summary, the solution to the integral involving the square root of e^x is to use u=e^-x/2 to get du=-\frac{1}{2}e^{-x/2}\cdot dx. This can be integrated by substitution u=cosy to get 2(\sqrt{1-e^x})-2\tanh^{-1}(\sqrt{1-e^x})+c.
  • #1
Schrodinger's Dog
835
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[SOLVED] Re: Integral involving square root of e^x

Homework Statement



[tex]\int \sqrt{1-e^{-x}}[/tex]

Homework Equations



Sub rule.

The Attempt at a Solution



I realized that it's fairly obvious I can use u=e^-x/2 to give [itex]\sqrt {1-u^2}[/itex]

but I'm kind of looking at the answers and I'm not seeing how I can get to them from here?

The answer I have from my maths program looks a bit awkward? Not sure how they got here?

[tex]2(\sqrt{1-e^x})-2\tanh^{-1}(\sqrt{1-e^x})+c[/tex]

Are there any other useful subs anyone can think of, or tips for this one.

Not a homework question as such but I thought this was the best place for it.
 
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  • #2
Well, remember that the actual interview you are trying to determine is
[tex]\int \sqrt{1-e^{-x}}dx[/tex]. Performing the substitution [itex]u=e^{-x/2}[/itex] gives [itex]du=-\frac{1}{2}e^{-x/2}\cdot dx[/itex], yielding[tex]-2\int\frac{\sqrt{1-u^2}}{u}\cdot du[/tex]. You can integrate this by another subsitution, u=cosy, say.
 
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  • #3
Thanks Christo I'm still not sure here. I'm sure I'm just being thick but could you spell it out, I'm not seeing how I'm going to end up with something even remotely like that answer, even with your sub? It's probably the hyperbolic function, I've got to admit they weren't heavily dealt with in the course I did. I can see where the answer turns into what it is with the sub cos y but the arctanh, where does that come from exactly?
 
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  • #4
Thanks Christo I'm still not sure here. I'm sure I'm just being thick but could you spell it out, I'm not seeing how I'm going to end up like that answer, even with your sub? It's probably the hyperbolic function, I've got to admit they weren't heavily dealt with in the course I did. I can see where the answer turns into what it is with the sub cos y but the arctanh, where does that come from exactly? Is it some clever trig identity I'm missing

I can see where you get

[tex]2(1-e^x)-2tanh^{-1}(1-e^x)+c[/tex]

from

[tex]-2\int\frac{\sqrt{1-u^2}}{u}\cdot du[/tex] but why the arctanh?
 
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  • #5
Hmm, I don't really know to be honest. My substitution doesn't really work, since the fraction has a "u" in the denominator and not a "u^2," which is what I must have assumed.

I guess we'll have to wait for someone else to come along, as I can't see how to integrate that by hand, at the moment.
 
  • #6
Hi Schrodinger's Dog! :smile:

Put u = sechv, du = -sechvtanhvdv.

Then ∫(√(1 - u^2))/udu = -∫(tanh^2(v)sechv/sechv)dy

= ∫(sech^2(v) - 1)dv

= tanhv - v

= √(1 - u^2)) - tanh^-1√(1 - u^2)

= √(1 - e^-x) - tanh^-1√(1 - e^-x). :smile:

(hmm … might have been quicker to start by putting e^(-x/2) = sechv :rolleyes:)
 
  • #7
Thanks tiny tim, when I get a spare moment I'll go through that thoroughly. :smile:
 

1. What is the formula for calculating the integral involving square root of e^x?

The formula for calculating the integral of square root of e^x is ∫√e^x dx = 2√e^x + C, where C is the constant of integration.

2. How do you solve an integral involving square root of e^x?

To solve an integral involving square root of e^x, you can use the formula ∫√e^x dx = 2√e^x + C and then integrate using the power rule or u-substitution method.

3. What is the significance of the constant of integration in the integral involving square root of e^x?

The constant of integration, denoted by C, is added to the solution of an integral to account for all possible solutions. It represents the unknown value that is lost during the process of differentiation and is necessary to include in the final answer.

4. Can you use integration by parts to solve an integral involving square root of e^x?

Yes, you can use integration by parts to solve an integral involving square root of e^x. This method involves breaking down the integral into two parts and then using the product rule of differentiation to solve it.

5. Are there any real-world applications of integrals involving square root of e^x?

Yes, integrals involving square root of e^x have various real-world applications in fields such as physics, engineering, and economics. For example, the integral can be used to calculate the work done by a variable force or to determine the rate of change in a population growth model.

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