In third-year astronomy, I’ve steered the homework straight into integrals of the form

$$\int u^2 e^{-u^2}\,du$$

which happily spits out an answer involving the error function $erf$. These integrals show up a lot in physics because the Gaussian function $e^{-u^2}$ shows up in physics theories all the time. This is because many physical processes fulfill the requirements of the Central Limit Theorem, which basically says “Get used to Gaussians, kids!”

Formally, the error function is

$$\mathrm{erf}(x) \equiv \frac{2}{\pi}\int_0^x e^{-t^2}\, dt.$$

You won’t be able to bust out any tricks from first-year calculus to determine the answer here. Instead, we define a function that has the value of the definite integral from a known bound. I like to cast this in terms of other functions like sine. You can’t immediately tell me what the sine of $7\pi/9$ is without reaching for a calculator. Instead, you’d punch this in and happily report its value (0.642788…). The error function should be treated the same way: punch in a number (the bound on the integral) and it happily reports the value of the integral. The only problem is that most calculators bury their error function capabilities, when they have it at all. Wolfram alpha will hand you the answer pretty quick, as will pretty much any numerical package.

This sort of thing becomes progressively more common as you work along in physics. Relatively few functions that show up in physics have well defined, closed form answers. In the end, we rely heavily on calculators and computers to develop numerical approximations to high precision. The math trick is to just define a function to be equal to the integral and call it solved. Or spend a career studying the properties of the now-defined function (like the Riemann zeta function).