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

Question Number 38720 by maxmathsup by imad last updated on 28/Jun/18

f i n d \int \frac{\sqrt{x + 1} - \sqrt{x - 1}}{\sqrt{x + 1} - \sqrt{x - 1}} d x

Commented by math khazana by abdo last updated on 28/Jun/18

t h e Q i s f i n d \int \frac{\sqrt{x + 1} - \sqrt{x - 1}}{\sqrt{x + 1} + \sqrt{x - 1}} d x

Commented by prof Abdo imad last updated on 29/Jun/18

I = \int \frac{{\sqrt{x + 1} - \sqrt{x - 1}}^{2}}{2} d x

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

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

= \frac{x^{2}}{2} - \int \sqrt{x^{2} - 1} d x b u t c h a n g e m e n t x = c h (t)

g i v e \int \sqrt{x^{2} - 1} d x = \int s h t s h t d t

= \int {s h}^{2} t d t = \int \frac{c h (2 t) - 1}{2} d t

= \frac{1}{4} s h (2 t) - \frac{t}{2} = \frac{1}{4} 2 s h (t) c h (t) - \frac{t}{2}

= \frac{1}{2} x \sqrt{x^{2} - 1} - \frac{1}{2} a r g c h (x)

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

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