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Question
Integrate the following with respect to x:\
`logx/(1 + log)^2`
Solution
I = `int logx/(1 + log x)^2 "d"x`
Put t = ` log x`
⇒ x = et
dx = et dt
I = `int "t"/(1 + "t")^2 "e"^"t" * "dt"`
I = `int "e"^"t" (1 + "t" - 1)/(1 + "t")^2 "dt"`
I = `int "e"^"t" [(1 + "t")/(1 + "t")^2 - 1/(1 + "t")^2] "dt"`
I = `int "e"^"t" [1/(1 + "t") - 1/(1 + "t")^2] "dt"`
f(t) = `1/(1 + "t")`
f'(t) = `- 1/(1 + "t")^2`
I = `"e"^"t" * 1/(1 + "t") + "c"`
= `x * 1/(1 + log x) + "c"`
I = `x/(1 + log x) + "c"`
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