prove-0-1-ln-4-2x-x-2-dx-2ln-2-e-pi-3-

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 . .