What is primitive of function 1/(sinx)^4?

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Let F be the primitive function

`F'=1/(sin(x))^4`

`F'=cosec^4(x)`

`F=intcosec^4(x)dx`

`=int cosec^(x) cosec^2(x)dx`

`=-cot(x)cosec^2(x)-int(2cosec(x)(-cosec(x)cot(x)))(-cot(x))dx`

`=-cot(x)cosec^2(x)-int(2cosec^2(x)cot^2(x))dx`

`=-cot(x)cosec^2(x)-2int(coesc^2(x)(cosec^2(x)-1))dx`

`=-cot(x)cosec^2(x)-2intcosec^4(x)dx+2intcosec^2(x)dx`

`=-cot(x)cosec^2(x)-2F-2cot(x)`

`3F=-cot(x)(cosec^2(x)+2)`

`F=-(1/3)cot(x)(cosec^2(x)+2)`

Thus primitive function is -(1/3)cot(x)(cosec^2(x)+2)+c , where c is independent of x i.e. constant .

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