find ∫ ((x^2 dx)/(x^3 −2x+1))


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Question Number 133123 by abdomsup last updated on 19/Feb/21

$f i n d \int \frac{x^{2} d x}{x^{3} - 2 x + 1}$
	Answered by Ñï= last updated on 19/Feb/21

	$\int \frac{x^{2}}{x^{3} - 2 x + 1} d x$ $= \frac{1}{3} \int \frac{3 x^{2} - 2 + 2}{x^{3} - 2 x + 1} d x$ $= \frac{1}{3} l n ∣ x^{3} - 2 x + 1 ∣ + \frac{2}{3} \int \frac{d x}{(x - 1) (x^{2} + x - 1)}$ $= \frac{1}{3} l n ∣ x^{3} - 2 x + 1 ∣ + \frac{2}{3} \int (\frac{1}{x - 1} - \frac{x + 2}{x^{2} + x - 1}) d x$ $= \frac{1}{3} l n ∣ x^{3} - 2 x + 1 ∣ + \frac{2}{3} l n ∣ x - 1 ∣ - \frac{1}{3} \int \frac{2 x + 1 + 3}{x^{2} + x - 1} d x$ $= \frac{1}{3} l n ∣ x^{3} - 2 x + 1 ∣ + \frac{2}{3} l n ∣ x - 1 ∣ - \frac{1}{3} l n ∣ x^{2} + x - 1 ∣ - \frac{1}{\sqrt{5}} l n ∣ \frac{x + \frac{1}{2} - \sqrt{\frac{5}{4}}}{x + \frac{1}{2} + \sqrt{\frac{5}{4}}} ∣ + C$
	Commented by mathmax by abdo last updated on 19/Feb/21

	$thNks sir$
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