Are there other kinds of bump functions than $e^\frac1{x^2-1}$? I've only seen the bump function $e^\frac1{x^2-1}$ so far.  Where could I find examples of functions $C^∞$ on $\mathbb{R}$ that are zero everywhere except on $(-1,1)$?

Are there others that do not involve the exponential function?  Are there any with a closed form integral?  Is there a preferred function?
There is even simpler example from this book of Loring Tu.

You simply start with
$$f(t)=\left\{\begin{array}{lr}
0&amp;t\leqslant 0\\
e^{-1/t}&amp;t&gt;0
\end{array}\right..$$
Then you define
$$g(t)=\frac{f(t)}{f(t)+f(1-t)}$$
and shift it to the right by creating
$$h(t)=g(t-1).$$
To make it symmetric you put
$$k(t)=h(t^2)$$
and finally to make it look like bump function you put
$$\rho(t)=1-k(t).$$
It will look like this

$\hspace{4,5cm}$


As you can tell from the picture, Loring Tu covered more general case (he even generalised this construction to smooth manifolds of arbitrary dimension).

For more details and picutres go to Tus book.



BONUS I created plots of functions $f,g,h,k,\rho$ in this geogebra sheet. Enjoy.
Seems to me that if you take the second derivative of a bump function you get another bump function, which should be not so hard to show. In fact all even derivatives are bump functions. Then you linearly transform them (shift in the x or y axis and/or rescale the axes) to fit your criteria.