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Question
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Options
`log ("cosec"x - cotx) + tan(x/2) + c`
sin 2x – cos x + c
`log (secx + tanx) - cot(x/2) + c`
cos 2x – sin x + c
Solution
`log (secx + tanx) - cot(x/2) + c`
[ Hint : `int 1/(cosx - cos^2x)*dx`
= `int 1/(cosx(1 - cosx))*dx`
= `int ((1 - cosx) + cosx)/(cosx(1 - cosx))*dx`
= `int (1/cosx + 1/(1 - cosx))*dx`
= `int [sec x + 1/2 "cosec"^2(x/2)]*dx`
= `log|secx + tanx|1/2((-cotx/2))/(1/2) + c`
= `log|secx + tanx| - cot(x/2) + c`].
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