please solve it integrate with respect to x ∫((5x−2)/(3x^2 +2x+1))


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Question Number 20167 by tammi last updated on 23/Aug/17

$p l e a s e s o l v e i t$ $i n t e g r a t e w i t h r e s p e c t t o x$ $\int \frac{5 x - 2}{3 x^{2} + 2 x + 1}$
	Answered by ajfour last updated on 23/Aug/17

	$= \frac{1}{3} \int \frac{(5 / 2) (2 x + 2 / 3) - 11 / 3}{{(x + 1 / 3)}^{2} + {(\sqrt{2} / 3)}^{2}} d x$ $= \frac{1}{3} \times \frac{5}{2} \int \frac{2 (x + 1 / 3)}{{(x + 1 / 3)}^{2} + {(\sqrt{2} / 3)}^{2}} d x$ $- \frac{11}{9} \int \frac{d x}{{(x + 1 / 3)}^{2} + {(\sqrt{2} / 3)}^{2}}$ $= \frac{5}{6} \ln (x^{2} + \frac{2 x}{3} + \frac{1}{3}) - \frac{11}{9} \times \frac{3}{\sqrt{2}} \tan^{- 1} (\frac{3 x + 1}{\sqrt{2}}) + C$ $= \frac{5}{6} \ln (3 x^{2} + 2 x + 1) - \frac{11}{3 \sqrt{2}} \tan^{- 1} (\frac{3 x + 1}{\sqrt{2}}) + C^{'} .$
	Commented by tammi last updated on 23/Aug/17

	$t h a n k s s s$
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