. What is the antiderivative of e^arctan(x)?

[silverdiesel]

What is the antiderivative of e^arctan(x)?
I can't seen to find this anywhere, although I suspect the answer would be pretty easy. This is the first problem of the first assignment in Calc 2. It is suposed to be review from Calc 1, but I can't figure it out.

 

[StatusX]

You can turn that into the integral of tan(u) e^u, which doesn't have an integral expressible in terms of elementary functions, and so neither does your original function. Now if that was arcsin or arccos in the exponent, it'd be a different story.

 

[d_leet]

You can turn that into the integral of tan(u) e^u, which doesn't have an integral expressible in terms of elementary functions, and so neither does your original function. Now if that was arcsin or arccos in the exponent, it'd be a different story.

You're subvstitution is wrong, it would work if he had xe^arctan(x) but you set u = arctan(x) which means x = tan(u) but there is no multiplication between x and e in his problem so you can get tan(u) e^u, but you could get sec²u e^udu.

 

[StatusX]

Right, and then I integrated by parts.

 

[d_leet]

Right, and then I integrated by parts.

Ah, yeah I see that now sorry about that.

 

[silverdiesel]

Thanks for your help. I did finally figure out the problem. It was a trick question, and I was not supposed to be able to figure out the antiderivitave. The whole problem was this:

d/dx(S₀¹e^{arctan x})dx

that "S" is supposed to be an integral sign.


I eventually figured.. The integral of e^{arctan x}, however you figure it, would be a number, and then the derivative of the number is just zero. I am still curious about the antiderivative... is there nothing that you can take the derivative of, to get e^{arctan x}?

 

[StatusX]

Unfortunately, most functions do not have "nice" antiderivatives. The classic example is the error function:

$$\mbox{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt$$

If we define elementary functions as exponentials, logs, algebraic functions (like polynomials and root extractions), and finite combinations (sums, products, etc) and compositions thereof (which, extending to complex numbers, allows you to include trigonometric functions and their inverses), then it can be proven there is no elementary function equivalent to the error function, nor to the antiderivative you've given here.

 

[silverdiesel]

hmm... interesting. I can't wait to learn all this.

one more question:

how do you write all the math symbols into these posts?

 

[StatusX]

It's called latex. https://www.physicsforums.com/showthread.php?t=8997" a good introduction to it.

 

[silverdiesel]

very cool, thanks so much StatusX, you have been very helpful.