find-dx-x-2-a-with-a-C-

Question Number 34237 by abdo mathsup 649 cc last updated on 03/May/18

f i n d \int \frac{d x}{x^{2} - a} w i t h a \in C .

Commented by Joel578 last updated on 04/May/18

t h a n k y o u f o r c o r r e c t i o n S i r

Commented by abdo mathsup 649 cc last updated on 04/May/18

l e t p u t \sqrt{a} = α + i β w i t h (α, β) \in R^{2}

I = \frac{1}{2 \sqrt{a}} \int (\frac{1}{x - \sqrt{a}} - \frac{1}{x + \sqrt{a}}) d x

= \frac{1}{2 (α + i β)} (\int \frac{d x}{x - \sqrt{a}} - \int \frac{d x}{x + \sqrt{a}}) b u t

\int \frac{d x}{x - \sqrt{a}} = \int \frac{d x}{x - α - i β}

= \int \frac{x - α + i β}{{(x - α)}^{2} + β^{2}} d x = \int \frac{x - α}{{(x - α)}^{2} + β^{2}} d x

+ i β \int \frac{d x}{{(x - α)}^{2} + β^{2}} b u t w e h a v e

\int \frac{x - α}{{(x - α)}^{2} + β^{2}} d x = \frac{1}{2} l n ({(x - α)}^{2} + β^{2}) + c_{1}

c h a n g e m e n t x - α = β t g i v e

\int \frac{d x}{{(x - α)}^{2} + β^{2}} = \int \frac{β d t}{β^{2} (1 + t^{2})} = \frac{1}{β} a r c t a n t + c_{2}

\Rightarrow \int \frac{d x}{x - \sqrt{a}} = \frac{1}{2} l n ({(x - α)}^{2} + β^{2}) + i a r c t a n t + λ

w e f o l o w t h e s a m e m e t h o d t o f i n d \int \frac{d x}{x + \sqrt{a}} \dots

Commented by math khazana by abdo last updated on 04/May/18

\int \frac{d x}{{(x - α)}^{2} + β^{2}} = \frac{1}{β} a r c t a n (\frac{x - α}{β}) + c_{2} \Rightarrow

\int \frac{d x}{x - \sqrt{a}} = \frac{1}{2} l n ({(x - α)}^{2} + β^{2}) + i a r c t a n (\frac{x - α}{β}) + c

Commented by abdo mathsup 649 cc last updated on 04/May/18

n v e r m i n d s i r j o e l .