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Question
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Options
`x cot (x/2) + c`
`- x cot (x/2) + c`
`cot (x/2) + c`
`x tan (x/2) + c`
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Solution
`- x cot (x/2) + c`
[ Hint : `int (x- sinx)/(1 - cosx)*dx = int (x - 2sin(x/2)cos(x/2))/(2sin^2 (x/2))*dx`
= `(1)/(2) int x"cosec"^2(x/2)*dx - int cot(x/2)*dx`
= `(1)/(2) [x int "cosec"^2 (x/2)*dx - int [d/dx(x) int "cosec"^2(x/2)^(dx)]*dx - int cot(x/2)*dx`
= `(1)/(2)[x{(-cot(x/2))/((1/2))} - int1* (-cot(x/2))/((1/2))*dx - intcot(x/2)*dx`
= `xcot(x/2) + int cot(x/2)*dx - int cot(x/2)*dx`
= `- x cot(x/2) + c`].
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