TITLE: How can a function have an antiderivative that can't be written?
QUESTION [11 upvotes]: If a function's integral can't be written, then how can we find exact values for it over areas? Can we only ever estimate it? Why can't we make new functions to define these strange unwritable anti-derivatives?

REPLY [34 votes]: We can, and do all the time! For example, 
The Gamma function
$$
\Gamma (z) = \int_0^{\infty} t^{z-1}\mathrm{e}^{-t} \ \mathrm{d}t.
$$
The Beta function 
$$
\mathrm{B}(z,y) = \int_0^1 t^{z-1}(1-t)^{y-1}\,\mathrm{d}t.
$$
The Exponential integral function
$$
\mathrm{E}_1(z) = \int_z^\infty \frac{e^{-t}}{t}\,  \mathrm{d}t.
$$
The Error function
$$
\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt.$$
The Elliptic integral of the second kind
$$
 E(\phi,k)=\int_0^{\phi} \sqrt{1-k^2\sin^2\theta} \ \mathrm{d}\theta. 
$$
The Logarithmic integral function
$$
 {\rm Li} (x) = \int_2^x \frac{\mathrm{d}t}{\ln t},
$$
and many, many more very important "special functions' are defined by definite integrals. If you go back even to elementary functions, you can define the logarithm via the following integral - 
The Logarithm
$$
\ln (t) = \int_1^t \frac{1}{x} \, dx.
$$
As to your first question, how do we find areas under these curves if we don't have an elementary antiderivative? Well, how do you find the area under the curve $1/t$ from $1$ to $5$? The above integral tells you the value is $\ln 5$, but what is that value, exactly? We can only approximate it, given the best methods we have! 
The same is true of all the above functions. At some special values they have exact values, given perhaps by integers, rational or irrational numbers, or a combination of common mathematical constants such as $\pi,\mathrm{e},\gamma,$ Catalan's constant, etc. (another interesting question is - why are these constants special enough to have names? Because they come up all the time! The same is true for the above functions) .
But for almost all values we must approximate the value of the function by computing the definite integral numerically (Trapezoid rule, Simpson's rule, more advanced techniques), or using some other representation of the integral such as an infinite sum etc.