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

Question Number 122205 by mathmax by abdo last updated on 14/Nov/20

find \int \frac{dx}{x (x + 1) \sqrt{x^{2} + x}}

Commented by liberty last updated on 15/Nov/20

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

= \int \frac{x^{- 2} dx}{x {(1 + x^{- 1})}^{\frac{3}{2}}}; [u^{\frac{2}{3}} = 1 + \frac{1}{x}]

\Rightarrow [\frac{2}{3} u^{- \frac{1}{3}} du = - x^{- 2} dx]

J = \int (u^{\frac{2}{3}} - 1) (- \frac{2}{3} u^{- \frac{1}{3}} du) (\frac{1}{u})

J = - \frac{2}{3} \int (u^{\frac{2}{3}} - 1) (u^{- \frac{4}{3}}) du

J = - \frac{2}{3} \int (u^{- \frac{2}{3}} - u^{- \frac{4}{3}}) du

J = - \frac{2}{3} [{3 u}^{\frac{1}{3}} + {3 u}^{- \frac{1}{3}}] + c

J = - 2 \sqrt{\frac{x + 1}{x}} - 2 \sqrt{\frac{x}{x + 1}} + c

J = - 2 (\frac{x + 1 + x}{\sqrt{x^{2} + x}}) + c = - \frac{4 x + 2}{\sqrt{x^{2} + x}} + c

Commented by benjo_mathlover last updated on 15/Nov/20

l e t J (x) = - \frac{4 x + 2}{\sqrt{x^{2} + x}}

\frac{d J (x)}{d x} = - [\frac{4 \sqrt{x^{2} + x} - (4 x + 2) (\frac{2 x + 1}{2 \sqrt{x^{2} + x}})}{x^{2} + x}]

= - [\frac{4 \sqrt{x^{2} + x} - \frac{{(2 x + 1)}^{2}}{\sqrt{x^{2} + x}}}{x^{2} + x}]

= - [\frac{4 x^{2} + 4 x - 4 x^{2} - 4 x - 1}{(x^{2} + x) \sqrt{x^{2} + x}}]

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

c o r r e c t s i r L i b e r t y .

Answered by TANMAY PANACEA last updated on 14/Nov/20

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

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

a = \frac{1}{x} \to d x = \frac{- 1}{a^{2}} d a b = \frac{1}{x + 1} \to d x = - \frac{1}{b^{2}} d b

\int \frac{- d a}{a^{2} \times \frac{1}{a} \times \sqrt{\frac{1}{a^{2}} + \frac{1}{a}}} - \int \frac{- d b}{b^{2} \times \frac{1}{b} \sqrt{{(\frac{1}{b} - 1)}^{2} + (\frac{1}{b} - 1)}}

\int \frac{- d a}{\sqrt{1 + a^{2}}} - \int \frac{d b}{b \times \sqrt{\frac{1}{b^{2}} - \frac{2}{b} + 1 + \frac{1}{b} - 1}}

\int \frac{- d a}{\sqrt{1 + a^{2}}} - \int \frac{d b}{\sqrt{1 - b}}

\int \frac{- d a}{\sqrt{1 + a^{2}}} + \int \frac{d (1 - b)}{\sqrt{1 - b}}

- l n (a + \sqrt{1 + a^{2}}) + \frac{{(1 - b)}^{\frac{1}{2}}}{\frac{1}{2}} + C

- l n (\frac{1}{x} + \sqrt{1 + \frac{1}{x^{2}}} + 2 {(1 - \frac{1}{x + 1})}^{\frac{1}{2}} + c