Solve ∫cos^-x/e^x^x? - Get Help Here

[Superposed_Cat]

∫cos^-x/e^x^x? can this be done? A thanks to anyone who can do this.

 

[D H]

What you posted makes *no* sense. It's a poorly formed formula. So of course it can't be done.

 

[pwsnafu]

What you wrote doesn't make sense. Cosine to the power of what? I'm assuming the variable of integration is x? What does e^x^x mean: ${e^x}^x$ or $(e^x)^x$?
And what does "can this be done" mean? Are you asking if it's integrable? Are you asking if it has an elementary derivative?

 

[Superposed_Cat]

x is variable of integration. its cos to the power of -x. and to pwsnafu. the former.

sorry

And what does "can this be done" mean? Are you asking if it's integrable? Are you asking if it has an elementary derivative?

yes.

 

[D H]

Cosine of what to the power of -x, Superposed_Cat?Suppose you do come up with a well-formed formula that bears some resemblance to what you wrote in the original post. Given the mess in the OP, it's almost assured not going to be integrable in terms of the elementary functions.

Example: What is ∫e^-x-2dx ? What about ∫sin(t)/t dt? These are of a much simpler form than what you wrote, and yet one cannot express these indefinite integrals as a closed form expression in the elementary functions.

 

[Superposed_Cat]

oh damn! I am forgetting so many things deepest apologies

 

[arildno]

Who cares?
It is a continuous function, and therefore integrable (wherever the numerator is well defined).
It does not have an antiderivative expressible in elementary functions, but over any finite interval, the value of the integral can readily be calculated by numerical means.

 

[Superposed_Cat]

Damn, I hoping there was an antiderivative.

 

[arildno]

Damn, I hoping there was an antiderivative.

It sure exists an antiderivative. It's just that is impossible to find a neat formula for it.

 

[R136a1]

It sure exists an antiderivative. It's just that is impossible to find a neat formula for it.

It all depends on what the OP is happy with, of course. Maybe he's happy by just being able to find the graph and a numerical way of computing things. After all, we don't really know more than that about logarithms or sines and cosines either. But still we interpret those functions as well-known.

On the other hand, if you want to express it as known elementary functions, then such a thing is probably impossible. But an antiderivative is certainly there.

So, it is up to the OP to clarify what he means with "finding" the integral.

 

[pwsnafu]

Damn, I hoping there was an antiderivative.

If f(x) is an integrable function, then $g(x) = \int_{a}^{x} f(t) \, dt$ is an antiderivative. This is why I asked what you mean by "can be done".

 

[arildno]

If f(x) is an integrable function, then $g(x) = \int_{a}^{x} f(t) \, dt$ is an antiderivative. This is why I asked what you mean by "can be done".

I strenuously oppose that that is a NEAT formula.
It is a voracious dragon in a mouse's clothing.

 

[D H]

Damn, I hoping there was an antiderivative.

Of course there's an antiderivative. The problem is that you can't express it in terms of a finite number of operations using only the elementary functions. You can express it, for example, as some kind of infinite series. Good luck developing that, though.

Another issue: that your function involves x^x (e^xx) and cos^-x(x) means there's a branch point at x=0. This is going to make the series have a finite radius of convergence. But an antiderivative certainly does exist.

 

[Superposed_Cat]

Damn, I hoping there was an antiderivative.

Translation:Damn, I was hoping that there was an antiderivative that could be found non-numerically in less than an hour.

 

[D H]

The answer then is no.

Why would you think anything there is?

 

[HallsofIvy]

Translation:Damn, I was hoping that there was an antiderivative that could be found non-numerically in less than an hour.

The answer to that would be "no" for almost every integrable function.

 

[arildno]

As HallsofIvy implies, what you learn in your studies amounts to practically..Nothing.