Evaluate-x-1-1-x-x-2-dx-

Question Number 50 by surabhi last updated on 25/Jan/15

Evaluate \int (x + 1) \sqrt{1 - x - x^{2}} d x .

Commented by 123456 last updated on 13/Dec/14

tente ? completar quadrados e fca uma substutuico

Answered by 123456 last updated on 13/Dec/14

hint :

- x^{2} - x + 1 =

- (x^{2} + x) + 1 =

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

- (x^{2} + 2 \times \frac{1}{2} \times x + \frac{1}{4} - \frac{1}{4}) + 1 =

- {(x + \frac{1}{2})}^{2} + \frac{5}{4} =

\frac{- {(2 x + 1)}^{2} + 5}{4}

from here you can use trigonometric substituition .

for references the answer is at the coments, good luck : 3

Commented by 123456 last updated on 13/Dec/14

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

\frac{1}{24} \sqrt{1 - x - x^{2}} (8 x^{2} + 14 x - 5) - \frac{5}{16} \sin^{- 1} (- \frac{2 x + 1}{\sqrt{5}}) + C

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