`int_0^(pi/4)2tan^3xdx=1-log2`

Explanation

`"Let i "= int_(0)^(pi//4) 2 tan^(3) x dx`
` =2 int_(0)^(pi//4) tan^(2) x. tan x dx`
` =2int_(0)^(pi//4) (sec^(2) x-1 )tan x dx`
` =2[int_(0)^(pi//4) sec^(2) x tan x dx -int_(0)^(pi//4) tan x dx]`
`=2int_(0)^(pi//4) (tanx )sec^(2) x dx -2[-log |cos x|]_(0)^(pi//4)`
` =2[(tan^(2))/(2)]_(0)^(pi//4) +2 [log |cos.(pi)/(4)|-log |cos 0|] =tan^(2) ((pi)/(4))-0+2 [log ((1)/(sqrt(2))) -log 1]`
`=1+2log 2^(-1//2) -0`
`=1 -2xx (1)/(2) log 2=1-log 2`
hence proved