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Question
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
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Solution
Let I = `int ("e"^x*log(sin"e"^x))/(tan("e"^x)) "d"x`
= `int"e"^x*log(sin"e"^x)*cot("e"^x) "d"x`
Put log(sin ex) = t
Differentiating w.r.t. x, we get
`(cos"e"^x)/(sin "e"^x) * "e"^x "d"x` = dt
∴ `"e"^x * cot "e"^x "d"x` = dt
∴ I = `int "t"* "dt" = "t"^2/2 + "c"`
∴ I = `([log(sin "e"^x)]^2)/2 + "c"`
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