1-2-1-4-x-x-1-1-1-x-2-1-x-1-2-dx-

Question Number 129014 by bramlexs22 last updated on 12/Jan/21

\int_{- \frac{1}{2}}^{- \frac{1}{4}} x (x + 1) \sqrt{1 + \frac{1}{x^{2}} + \frac{1}{{(x + 1)}^{2}}} d x = ?

Commented by Ajao yinka last updated on 12/Jan/21

37/192

Answered by liberty last updated on 12/Jan/21

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

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

\sqrt{x^{2} {(x + 1)}^{2}} = ∣ x (x + 1) ∣ = - x (x + 1) for - \frac{1}{4} ⩽ x ⩽ - \frac{1}{2}

\sqrt{x^{4} + {2 x}^{3} + {3 x}^{2} + 2 x + 1} = \sqrt{{(x^{2} + x + 1)}^{2}}

= ∣ x^{2} + x + 1 ∣ > 0 for - \frac{1}{2} ⩽ x ⩽ - \frac{1}{4}

Answered by Dwaipayan Shikari last updated on 12/Jan/21

\int_{- \frac{1}{2}}^{- \frac{1}{4}} ∣ x^{2} + x + 1 ∣ d x

= - (\frac{7}{192} - \frac{3}{32} + \frac{1}{4}) = - \frac{37}{192}