find-x-2-dx-x-3-2x-1-

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