integration-x-3-x-2-x-1-

Question Number 28701 by students last updated on 29/Jan/18

i n t e g r a t i o n \frac{x^{3}}{x^{2} + x + 1}

Commented by abdo imad last updated on 29/Jan/18

\int \frac{x^{3}}{x^{2} + x + 1} d x = \int \frac{x^{3} - 1 + 1}{x^{2} + x + 1} d x = \int (x - 1) d x + + \int \frac{d x}{x^{2} + x + 1}

= \frac{x^{2}}{2} - x + \int \frac{d x}{x^{2} + x + 1} b u t

I = \int \frac{d x}{x^{2} + x + 1} = \int \frac{d x}{{(x + \frac{1}{2})}^{2} + \frac{3}{4}} l e t u s e t h e c h . x + \frac{1}{2} = \frac{\sqrt{3}}{2} t

I = \int \frac{1}{\frac{3}{4} t^{2} + \frac{3}{4}} \frac{\sqrt{3}}{2} d t = \frac{\sqrt{3}}{2} \frac{4}{3} \int \frac{d t}{1 + t^{2}} = \frac{2 \sqrt{3}}{3} a r c t a n t

= \frac{2 \sqrt{3}}{3} a r c t a n (\frac{2}{\sqrt{3}} (x + \frac{1}{2})) s o

\int \frac{x^{3}}{x^{2} + x + 1} d x = \frac{x^{2}}{2} - x + \frac{2 \sqrt{3}}{3} a r c t a n (\frac{2}{\sqrt{3}} (x + \frac{1}{2})) .

Commented by abdo imad last updated on 29/Jan/18

\int (\dots) d x = \dots \dots . . + λ .

Answered by mrW2 last updated on 29/Jan/18

= \frac{x^{3} + x^{2} + x - x^{2} - x - 1 + 1}{x^{2} + x + 1}

= x - 1 + \frac{1}{x^{2} + x + 1}

= x - 1 + \frac{1}{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}}

\int \frac{x^{3}}{x^{2} + x + 1} d x

= \frac{x^{2}}{2} - x + \frac{2}{\sqrt{3}} \tan^{- 1} \frac{x + \frac{1}{2}}{\frac{\sqrt{3}}{2}} + C

= \frac{x^{2}}{2} - x + \frac{2}{\sqrt{3}} \tan^{- 1} \frac{2 x + 1}{\sqrt{3}} + C

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