prove:∫_0 ^( 1) ln (4− 2x +x^( 2) )dx =2ln((2/e)) +(π/( (√3)))


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Question Number 154469 by mnjuly1970 last updated on 18/Sep/21

$p r o v e : \int_{0}^{1} l n (4 - 2 x + x^{2}) d x = 2 l n (\frac{2}{e}) + \frac{π}{\sqrt{3}}$
	Answered by ARUNG_Brandon_MBU last updated on 18/Sep/21

	$I = \int_{0}^{1} \ln (4 - 2 x + x^{2}) d x = \int_{0}^{1} \ln (3 + {(1 - x)}^{2}) d x$ $= \int_{0}^{1} \ln (3 + x^{2}) d x = {[x \ln (3 + x^{2})]}_{0}^{1} - \int_{0}^{1} \frac{2 x^{2}}{x^{2} + 3} d x$ $= \ln 4 - 2 \int_{0}^{1} (1 - \frac{3}{x^{2} + 3}) d x = 2 \ln 2 - 2 + 2 \sqrt{3} {[\tan^{- 1} (\frac{x}{\sqrt{3}})]}_{0}^{1}$ $= 2 \ln 2 - 2 \ln e + \frac{\sqrt{3}}{3} π = 2 \ln (\frac{2}{e}) + \frac{π}{\sqrt{3}}$
	Commented by mnjuly1970 last updated on 18/Sep/21

	$g r a t e f u l . . v e r y n i c e s o l u t i o n . .$
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