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If inte^(x)((1-sinx)/(1-cosx))dx=f(x)+ ...

If `inte^(x)((1-sinx)/(1-cosx))dx=f(x)+` Constant, then f(x) is equal to

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We have `I=inte^(x)*((1-sinx)/(1-cosx))dx=inte^(x)*{(1)/((1-cosx))-(sinx)/((1-cosx))}dx`
`=inte^(x)*{(1)/(2sin^(2)(x//2))-(2sin(x//2)cos(x//2))/(2sin^(2)(x//2))}dx`
`=inte^(x)*{(1)/(2)"cosec"^(2)(x)/(2)-cot(x)/(2)}dx`
`=inte^(x)*{-cot(x)/(2)+(1)/(2)"cosec"^(2)(x)/(2)}dx`
`inte^(x)*{f(x)+f'(x)}dx,"where"f(x)=-cot(x)/(2)andf'(x)=(1)/(2)"cosec"^(2)(x)/(2)`
`e^(x)*f(x)+C=e^(x)(-"cot"(x)/(2))+C=-e^(x)"cot"(x)/(2)+C`.
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