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Question
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
Options
`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"`
Solution
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to `"cosec"("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C"`.
Explanation:
Let I = `int "dx"/(sin(x - "a")sin(x - "b"))`
Multiplying and dividing by sin(b – a) we get,
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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