1. Jan 21, 2006

### silverdiesel

What is the antiderivative of earctan(x)?
I cant 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 cant figure it out.:surprised :grumpy: :yuck:

2. Jan 21, 2006

### StatusX

You can turn that into the integral of tan(u) eu, 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.

Last edited: Jan 21, 2006
3. Jan 21, 2006

### d_leet

You're subvstitution is wrong, it would work if he had xearctan(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) eu, but you could get sec2u eudu.

4. Jan 21, 2006

### StatusX

Right, and then I integrated by parts.

5. Jan 21, 2006

### d_leet

Ah, yeah I see that now sorry about that.

6. Jan 21, 2006

### 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(S01earctan x)dx

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

I eventually figured.. The integral of earctan 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 earctan x?

7. Jan 21, 2006

### 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.

Last edited: Jan 21, 2006
8. Jan 21, 2006

### silverdiesel

hmm... interesting. I cant wait to learn all this.

one more question:

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

9. Jan 21, 2006

### StatusX

Last edited by a moderator: Apr 21, 2017
10. Jan 21, 2006

### silverdiesel

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