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Question Number 142344 by mnjuly1970 last updated on 30/May/21

$$$$$$\mathit{\varphi }:={\underset{\frac{1}{e}}{\int }}^e\{\frac{1}{ln\left(x\right)}+ln\left(ln\left(x\right)\right)\}dx$$

Commented by Dwaipayan Shikari last updated on 30/May/21

$$log\left(x\right)=t$$$$=\int (\frac{1}{t}+log\left(t\right))e^{t}dt=e^{t}log\left(t\right)+C=xlog\left(log\left(x\right)\right)+C$$$$Now\frac{d}{dt}\left(e^{t}f\left(t\right)\right)=e^t{f}^{\prime }\left(t\right)+e^{t}f\left(t\right)\Rightarrow e^{t}f\left(t\right)=\int e^t(f^{\prime }\left(t\right)+f\left(t\right))dt$$$$$$

Commented by rs4089 last updated on 30/May/21

$$but,whataboutlimit?$$

Commented by Dwaipayan Shikari last updated on 30/May/21

$$log\left(log\left(x\right)\right)isnotdefinedat\frac{1}{e}$$$$log\left(log\left(\frac{1}{e}\right)\right)=log(-1)=i\pi$$

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