All of the integrands in the Integration Bee contest are elementary functions, that is, finite combinations of algebraic, trigonometric, and exponential functions, and their inverses. So are all of their antiderivatives.
But the horrible truth is that many (in some sense most) elementary
functions
have antiderivatives that cannot be expressed in terms of elementary
functions.
For example, although you can find an explicit formula for an
antiderivative of
,
there is no explicit formula in terms of elementary functions for an
antiderivative of 
.
For information about why there is no such formula in general, see:
There are many simple-looking functions that have no elementary-function antiderivative. Such nonelementary integrals include:
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Since there is only a small lexicographical difference between , which
cannot be expressed in terms of elementary functions, and
,
which can easily be evaluated, you can see that antidifferentiation is
a touchy business. (All the
Integration Bee problems have been carefully rigged so that the art of
finding explicit antiderivatives can be successfully applied.)
The skill celebrated by the Integration Bee contest is thus not broadly applicable to indefinite integrals. In general, all we know about the integral of an elementary function is the assertion of the Fundamental Theorem of Calculus: Every continuous function has an antiderivative. (But there is no guarantee we can find a formula for an antiderivative in terms of elementary functions like sine, cosine, logarithm, and so forth.)