Question

What is the value of ∫_0^1▒ln⁡x/(1+ln⁡x )^2 dx

Answer 1

what is this computer language

Answer 2

I want to integrate (log x)/(1 + log x) between the limits 0 to 1

Answer 3

sorry (log x)/(1 + log x)^2

Answer 4

\[\frac{\ln x}{(1+\ln x)^2}\]

Answer 5

yes

Answer 6

there is a singularity in \(x=\frac{1}{e}\)

Answer 7

can we integrate it or not

Answer 8

we can...im searchin for it...one of the users asked it before

Answer 9

please tell me if u find it thanks

Answer 10

i cant find it ...but note that\[\int \frac{\ln x}{(1+\ln x)^2} \text{d}x=\frac{x}{1+\ln x}+C\]and play around this\[\int_{0}^{1} \frac{\ln x}{(1+\ln x)^2} \text{d}x=\lim_{\epsilon \rightarrow 0} (\int_{0}^{\frac{1}{e}-\epsilon} \frac{\ln x}{(1+\ln x)^2} \text{d}x+\int_{\frac{1}{e}+\epsilon}^{1} \frac{\ln x}{(1+\ln x)^2} \text{d}x)\]

Answer 11

@mukushla you going for Cauchy Principal value ?
i was thinking By parts .

Answer 12

thanks bt u give me direct answer can u explain it how it comes

Answer 13

bt i think we cannot solve it by parts

Answer 14

i think we can solve it by parts. Wolfram also says yes Parts can be used.
http://www.wolframalpha.com/input/?i=integrate+%28ln%28x%29%2F%281%2Bln%28x%29%29%5E2

Answer 15

thanks sami thank u very much