So one of the question on the MIT Integration bee has baffled me all day today ∫π40tan2x1+x2dx I have tried a variety of things to do this, starting with Integration By Parts Part 1 tanx−x1+x2|π40−∫π40−2x(tanx−x)(1+x2)2dx which that second integral is not promising, so then we try Integration By Parts Part 2 tan−1xtan2x|π40−∫π402tan−1xtanxsec2xdx which also does not seem promising Trig Substitution x=tanθ which results ∫tan−1π40tan2(tanθ)dθ which I think too simple to do anything with (which may or may not be a valid reason for stopping here) I had some ideas following this like power reducing tan2x=1−cos2x1+cos2x which didn't spawn any new ideas. Then I thought maybe something could be done with differentiation under the integral but I could not figure out how to incorporate that. I also considered something with symmetry somehow which availed no results. I'm also fairly certain no indefinite integral exists. Now the answer MIT gave was 13 but wolfram alpha gave ≈ .156503. Note The integral I gave was a simplified version of the original here is the original in case someone can do something with it ∫π401−x2+x4−x6...cos2x+cos4x+cos6x...dx My simplification is verifiably correct, I'd prefer no complex analysis and this is from this Youtube Video close to the end.
Answer
There must be some problem here. Note that on [0,π/4], tan2x1+x2≤tan2x1+02=tan2x, therefore the definite integral is bounded above as follows: 0≤∫π/4x=0tan2x1+x2dx<∫π/4x=0tan2xdx=1−π4<13. Mathematica is correct; the definite integral cannot be 1/3. I also watched the YouTube video (see time stamp 1:34:34), and you have transcribed the question correctly and performed the correct algebraic transformations.
Okay so at 1:39:04 they reveal the answer as 1/3. This is very, very obviously wrong. The very first thought to enter my mind was to check the reasonableness of the result by choosing an appropriate bound.
Interestingly, one contestant at 1:38:42 answers with π/16≈0.19635 which is remarkably accurate for a last-moment guess, certainly much closer to the mark than the official answer!
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