2-x-3-2-x-3-dx-

Question Number 82577 by M±th+et£s last updated on 22/Feb/20

\int \frac{2 - \sqrt{x + 3}}{2 + \sqrt{x - 3}} d x

Commented by mathmax by abdo last updated on 22/Feb/20

l e t I = \int \frac{2 - \sqrt{x + 3}}{2 + \sqrt{x + 3}} d x v h a n g e m e n t 2 + \sqrt{x + 3} = t g i v e

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

I = \int \frac{2 - \sqrt{t^{2} - 4 t + 1 + 3}}{t} (2 t - 4) d t = 2 \int \frac{(t - 2) \sqrt{t^{2} - 4 t + 4}}{t} d t

= 2 \int \frac{(t - 2) (t - 2)}{t} d t = 2 \int \frac{t^{2} - 4 t + 4}{t} d t = 2 \int (t - 4 + \frac{4}{t}) d t

= t^{2} - 8 t + 8 l n ∣ t ∣ + C

= {(2 + \sqrt{x + 3})}^{2} - 8 (2 + \sqrt{x + 3}) + 8 l n ∣ 2 + \sqrt{x + 3} ∣ + C

Commented by M±th+et£s last updated on 22/Feb/20

g o d b l e s s y o u s i r

Commented by msup trace by abdo last updated on 22/Feb/20

t h a n k y o u s i r .