1. integrate sin 2xd(tanx)

let tan x =t;
sin 2x= 2 sinxcosx
​divide and multiply by cos x on R.H.S.

sin 2x= 2tan x cos​2 x
sin 2x = 2tanx /{sec2x}
sin 2x = 2 tanx/{tan2x +1}
sin2x = 2t /( t21)                       as tan x =t;
so now you have to integrate..
{2t /( t21) }.dt

on solving it ,,we get 
log(t2+1)+C
or
log(tan2x+1)+C.
i think this is correct... :p
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Integration of sin 2x with the respect of tan x
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