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maths help with length of x^2 at 2 points

http://imgur.com/gallery/9KuZaRP
okay so here is the solution. what I want to know is how they got to 1/2*tanx = x. any help is appreciated
Reply 1
Avatar for nyxnko_
nyxnko_
20
Original post by SandeshGurung02
http://imgur.com/gallery/9KuZaRP
okay so here is the solution. what I want to know is how they got to 1/2*tanx = x. any help is appreciated


oh gotta love kumon :love:
I believe that is just a substitution that they chose to make life easier for themselves :yes: They could have chosen a different substitution but they made use of the identity tan⁡2θ+1≡sec⁡2θ\tan^2\theta + 1 \equiv \sec^2\theta to get to the next line, i guess :dontknow:
idk for sure but that's what i think. are there any other substitutions you would have used?
Original post by SandeshGurung02
http://imgur.com/gallery/9KuZaRP
okay so here is the solution. what I want to know is how they got to 1/2*tanx = x. any help is appreciated

I don't see "1/2*tan x = x" anywhere.
Reply 3
Avatar for FoxNuke
FoxNuke
10
Original post by SandeshGurung02
http://imgur.com/gallery/9KuZaRP
okay so here is the solution. what I want to know is how they got to 1/2*tanx = x. any help is appreciated

They want to get rid of the square root since it is easier to integrate without it.
But substituting x = tan(theta)/2,
√(1+4x²) = √(1+4(tan(t)/2)²) = √(1+4(tan²(t)/4)) = √(1+tan²(t)) = √sec²(t) = sec(t)

The general rule in picking these numbers depends on the radical.
√(a + bx²) makes the substitution x = √(a/b)tan(t) work. The end result is that you'll be able to factorise the √a out of the expression, and be left with a nice sec(t)

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