`int "cosec"^(3) x dx` का मान ज्ञात कीजिए।

Answer

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Explanation

माना `I=int cosec^(3) x dx =int cosec x. cosex^(2) x dx`
खण्डशः समाकलन करने पर
`I=cosec x. int cosec^(2) x dx -int [D(cosec x). Int cosec^(2) x dx] dx`
`=cosec x.(-cot x)-int (-cosec x cot x)(-cot x) dx`
`=-cosec x cot x-int cosec x cot^(2) x dx`
`=-cosec x cot x-int cosec x. (cosec^(2) x-1) dx`
`=-cosec x cot x - int cosec^(3) x dx + int cosec x dx`
`=-cosec x cot x-I+log |"tan" x/2|`
`:. 2I=-cosec x cot x+ log |"tan" x/2|`
`implies I=-1/2 cosec x cot x+1/2 log |"tan" x/2|+c`.