Question

Prove the following:

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

Answer 1

`int_0^(pi/4) 2 tan^3 x dx`

`= int_0^(pi/4) 2 tan x* tan^2 x  dx`

`= int_0^(pi/4) 2 tan x (sec^2 x - 1)  dx`

`= 2 int_0^(pi/4) (tan x) sec^2 x dx - 2 int_0^(pi/2) tan x dx`

`= 2 [(tan^2 x)/2]_0^(pi/4) - 2 [- log |cos x|] _0^(pi/4)`

`= (tan^2  pi/4 - tan^2 0) + 2 (log cos  pi/4 - log cos 0)`

`= (1 - 0) + 2 (log  1/ sqrt2 - log 1)`

`= 1 + 2 (log 1 - log sqrt 2 - log 1)`

`= 1 + 2 xx (-1/2 log 2)`

= 1 - log 2