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Question
Integrate the following with respect to x :
`("cosec" x)/(log(tan x/2))`
Solution
`int ("cosec" x)/(log(tan x/2)) "d"x`
Put `log (tan x/2)` = u
`1/(tan x/2) xx sec^2 x/2 xx 1/2 xx "d"x` = du
`cot x/2 xx 1/(cos^2 x/2) xx 1/2 xx "d"x` = du
`(cos x/2)/(sin x/2) xx 1/(cos^2 x/2) xx 1/2 xx "d"x` = du
`1/(2sin x/2 cos x/2) "d"x` = du
`1/sinx "d"x` = du
`"cosec" x "d"x` = du
∴ `int ("cosec" x)/(log(tan x/2)) "d"x = int "du"/"u"`
= `log |"u"| + "c"`
`int ("cosec" x)/(log(tan x/2)) "d"x = log |log(tan x/2)| + "c"`
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