When integrating and differentiating, sometimes one direction is easy and the other is harder. A nice example is $\frac{d}{dx}\tan x=\sec^2x$, where differentiating is easy but integration (without knowledge of the other direction) is difficult. So to get round the difficult direction you might integrate it thus: $$\begin{align*} \int\sec^2xdx&=?\\ u&=\tan x\\ du&=\sec^2xdx\\ \int\sec^2xdx&=\int du,\\ &=u+c\\ &=\tan x+c \end{align*}$$ This is correct. However, it leaves a bitter taste in my mouth, and I would be hesitant to award any marks for this in an exam. The issue is with the "oracle": the student already knows the answer.
My question is twofold.
- Should the above solution get full marks?
- If not, how can I explain this to my students (and also to my tutors)?
I am really struggling with (2) - I am struggling to verbalise my issues with this approach*. I am asking (1) because the struggle with (2) makes me wonder if my initial assumption is incorrect (that is, I wonder if the answer to (1) is "yes").
Also, I could phrase the exam question to specifically ban this approach. However, that is beside the point - I want to better understand why this approach should be banned!
*I chose to study maths all those years ago because I thought I wouldn't have to write much...have opinions...communicate...that sort of thing...