calculate ∫ ((lnx)/(x +x(lnx)^2 ))dx


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Question Number 42392 by abdo.msup.com last updated on 24/Aug/18

$c a l c u l a t e \int \frac{l n x}{x + x {(l n x)}^{2}} d x$
Commented by maxmathsup by imad last updated on 25/Aug/18

$c h a n g e m e n t l n (x) = t g i v e$ $I = \int \frac{t}{e^{t} + e^{t} t^{2}} e^{t} d t = \int \frac{t}{1 + t^{2}} d t = \frac{1}{2} l n (1 + t^{2}) + c$ $I = \frac{1}{2} l n (1 + {l n}^{2} (x)) + c .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 24/Aug/18
	$t=1+(lnx)^2 dt^ =((2lnx)/x)dx ∫((lnx)/(x{1+(lnx)^2 }))dx ∫(dt/(2(1+t^2 ))) (1/2)tan^(−1) (t)+c (1/2)tan^(−1) {1+(lnx)^2 [ +c$
	$t = 1 + {(l n x)}^{2} d \overset{}{t} = \frac{2 l n x}{x} d x$ $\int \frac{l n x}{x {1 + {(l n x)}^{2}}} d x$ $\int \frac{d t}{2 (1 + t^{2})}$ $\frac{1}{2} {t a n}^{- 1} (t) + c$ $\frac{1}{2} {t a n}^{- 1} {1 + {(l n x)}^{2} [+ c$
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