Evaluate: (i) `int{1+tanxtan(x+theta)}\ dx` (ii) `int(sin2x)/(sin(x-pi/6)sin(x+pi/6))\ dx`

Explanation

(i)

`text { Let I }=int 1+tan x tan (x+theta) d x `

`=int 1+tan x(frac{tan x+tan theta}{1-tan x tan theta}) d x `

`=int frac{1+tan ^{2} x}{1-tan x tan theta} d x `

`=int frac{sec ^{2} x d x}{1-tan x tan theta} `

`text { Putting tan } x=t `

`Rightarrow sec ^{2} x=frac{d t}{d x} `

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