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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.

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    $\begingroup$ See also: academia.stackexchange.com/questions/80898/… $\endgroup$ – Daniel R. Collins Jan 2 '17 at 0:22
  • $\begingroup$ I am reminded of the "near miss" $1782^{12} + 1841^{12} \stackrel{?}= 1922^{12}$. (Take the twelfth root of the left hand side in Mathematica/Wolfram Alpha and check its Approximate Form...) If I asked someone to show this was not true on a high school examination, I would hope for, e.g., "this cannot be the case because the LHS is odd but the RHS is even." If their stated reason was violating Fermat's Last Theorem, then I would ask them to explain the proof of FLT to me. And if they could do that, I would have to rethink their participation in my class... $\endgroup$ – Benjamin Dickman Jan 2 '17 at 0:36
  • $\begingroup$ @BenjaminDickman I am sadly in a high school that has no more courses... So if I could... $\endgroup$ – Simply Beautiful Art Jan 2 '17 at 1:10
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    $\begingroup$ @DanielR.Collins Perfect link! Thanks for that! $\endgroup$ – Simply Beautiful Art Jan 2 '17 at 1:14
  • $\begingroup$ Can you prove the integral formula, or have you only memorized it? Can you prove the the Gamma function has this connection with factorials? You should be able to prove the integration by parts formula, and so you should be able to to this integral in a completely elementary way. Not showing this elementary approach, and instead relying on a memorized formula, does not show mastery. $\endgroup$ – Steven Gubkin Jan 2 '17 at 15:26
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Quoted from the first part of my answer on https://academia.stackexchange.com/questions/80898/should-a-student-be-penalized-for-using-a-theorem-outside-of-the-curriculum/:

... the point of an exam is to assess mastery of basic knowledge covered in the course. If one uses a more powerful outside theorem, then the steps that they've skipped likely contained important concepts or techniques, that there is now no proof that the student has mastered. So the professor needs to ping the student to demonstrate those basic techniques before progressing.

More complete answer at the Academia SE link above.

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    $\begingroup$ I've added more context to the question, and if you could consider it, that'd be nice. $\endgroup$ – Simply Beautiful Art Jan 2 '17 at 1:19
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    $\begingroup$ @SimpleArt: The extra details wouldn't change my response any. $\endgroup$ – Daniel R. Collins Jan 2 '17 at 3:20

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