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Question
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
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Solution
Let I = `int ["cosec"(logx)][1 - cot(logx)] "d"x`
Put logex = t
∴ x = et
∴ dx = `"e"^"t"*"dt"`
∴ I = `int "cosec" "t"(1 - cot "t") "e"^"t" "dt"`
= `int "e"^"t" ("cosec" "t" - "cosec" "t"*cot "t") "dt"`
Put f(t) = cosec t
∴ f'(t) = −cosec t.cot t
∴ I = `int"e"^"t" ["f"("t") + "f'"("t")] "dt"`
= et ⋅ f(t) + c = et cosec t + c
∴ I = `x "cosec" (logx) + "c"`
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