I happen to be a student in America taking AP Calculus BC, or Calculus II, and recently, I had the following problem:
Determine whether the following integral converges and evaluate it if it does:
$$\int_0^\infty xe^{-x}\ dx$$
And my response:
$\int_0^\infty x^se^{-x}\ dx$ converges absolutely for $\text{Re}(s)>-1$ and $\int_0^\infty x^se^{-x}\ dx=\Gamma(s+1)\stackrel{s\in\mathbb N}=s!$.
Reference: en.wikipedia.org/wiki/Gamma_function
$\int_0^\infty xe^{-x}\ dx=1!=1$
This question happened to show up around the section of integration by parts and improper integrals, and as I used neither, I only got one point on this question for the apparently correct answer.
Of course, I knew this would most likely happen, but as it was, I wasn't particularly worried about getting it right or wrong since the teacher and I both know I am sufficient in my Calculus, but I was wondering what you guys though about this scenario.
The question is basically that an integral like this is "well-known" to me, and though I know the rubric like the back of my hand, I wonder what stance something well-known has when looking at a problem, like whether or not introducing special functions far beyond the scope of the course could/should be reasonable.
Indeed, as some more context, I am in the class only for credits and GPA and such, and I passed a mock final exam with flying colors.
For example, I post answers like this one all the time on MSE, so something like the above is fairly elementary for me.