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Question
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
Solution
`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
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