`int(x - sinx)/(1-cosx)dx` का मान ज्ञात कीजिए ।

Options

A. `x cot.(x)/(3)+c`
B. `-x cot.(x)/(3)+c`
C. `-x cot.(x)/(2)+c`
D. `x cot.(x)/(2)+c`

Answer

C

Explanation

`int(x-sinx)/(1-cosx)dx=int(x-2sin.(x)/(2)cos.(x)/(2))/(2sin^(2).(x)/(2))dx`
`=int(x)/(2sin^(2).(x)/(2))dx-int(2sin.(x)/(2)cos.(x)/(2))/(2sin^(2).(x)/(2))dx`
`=(1)/(2)int x" cosec"^(2)(x)/(2)dx- intcot.(x)/(2)dx`
`=(1)/(2)[x int" cosec"^(2)(x)/(2)dx-int{(d)/(dx)x int" cosec"^(2)(x)/(2)dx}dx]-intcot.(x)/(2)dx`
`=(1)/(2)[x.((-cot.(x)/(2)))/((1)/(2))-int1.((-cot.(x)/(2)))/((1)/(2))dx]-intcot.(x)/(2)+c`
`=-x cot.(x)/(2)+intcot.(x)/(2)dx- int cot.(x)/(2)dx+c`
`=-x cot.(x)/(2)+c`