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प्रश्न
`("d"x)/(sin (x - "a") sin (x - "b"))` बराबर है
पर्याय
`sin ("b" - "a") log |(sin(x - "b"))/(sin(x - "a"))| + "C"`
`"cosec" ("b" - "a") log |(sin(x - "a"))/(sin(x - "b"))| + "C"`
`"cosec" ("b" - "a") log |(sin(x - "b"))/(sin(x - "a"))| + "C"`
`sin ("b" - "a") log |(sin(x - "a"))/(sin(x - "b"))| + "C"`
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उत्तर
सही उत्तर `underline("cosec" ("b" - "a") log |(sin(x - "b"))/(sin(x - "a"))| + "C")` है।
व्याख्या:
मान लीजिए I = `int "dx"/(sin(x - "a")sin(x - "b"))`
हमें sin(b – a) से गुणा और भाग करना प्राप्त होता है
I = `1/(sin("b" - "a")) int (sin("b" - "a"))/(sin(x - "a") * sin(x - "b")) "d"x`
= `1/(sin("b" - "a")) int (sin(x + "b" - x - "a"))/(sin(x - "a") * sin(x - "b")) "d"x`
= `1/(sin("b" - "a")) int (sin[(x - "a") - (x - "b")])/(sin(x - "a") * sin(x - "b")) "d"x`
= `1/(sin("b" - "a")) int (sin(x - "a") cos(x - "b") - cos(x - "a") sin(x - "b"))/(sin(x - "a") * sin(x - "b")) "d"x`
= `1/(sin("b" - "a")) int (sin(x - "a") * cos(x - "b"))/(sin(x - "a")*sin(x - "b")) - (cos(x - "a")*sin(x - "b"))/(sin(x - "a") * sin(x - "b")) "d"x`
= `1/(sin("b" - "a")) int [(cos(x - "b"))/(sin(x - "b")) - (cos(x - "a"))/(sin(x - "a"))]"d"x`
= `1/(sin("b" - "a")) int [cot(x - "b") - cot(x - "a")]"d"x`
= `1/(sin("b" - "a")) [log sin(x - "b") - logsin(x - "a")] + "C"`
= `1/(sin("b" - "a")) * log|(sin(x - "b"))/(sin(x - "a"))| + "C"`
I = `"cosec"("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C"`.
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