∫ ((x−1)/(x+x^2 ln x)) dx ?


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Question Number 105574 by bemath last updated on 30/Jul/20

$\int \frac{x - 1}{x + x^{2} \ln x} d x ?$
	Answered by bobhans last updated on 30/Jul/20

	$s e t x = e^{u}, d x = e^{u} d u$ $I = \int \frac{(e^{u} - 1) e^{u}}{e^{u} + {u e}^{2 u}} d u = \int \frac{e^{u} - 1}{1 + {u e}^{u}} d u$ $I = \int (\frac{e^{u} + {u e}^{u}}{1 + {u e}^{u}} - 1) d u$ $s u b s t i t u t e v = 1 + {u e}^{u}$ $I = \int \frac{d v}{v} - \int d u = \ln v - u + c$ $I = \ln (1 + {u e}^{u}) - u + c$ $I = \ln (1 + x \ln x) - \ln x + c ★$ $I = \ln (\frac{1}{x} + \ln x) + c$
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