find-f-a-dx-1-ax-2-1-ax-2-with-a-gt-0-

Question Number 51991 by maxmathsup by imad last updated on 01/Jan/19

f i n d f (a) = \int \frac{d x}{\sqrt{1 + {a x}^{2}} + \sqrt{1 - {a x}^{2}}} w i t h a > 0

Commented by Abdo msup. last updated on 05/Jan/19

c h a n g e m e n t x \sqrt{a} = t g i v e

f (a) = \int \frac{d t}{\sqrt{a} (\sqrt{1 + t^{2}} + \sqrt{1 - t^{2}})}

= \frac{1}{\sqrt{a}} \int \frac{d t}{\sqrt{1 + t^{2}} + \sqrt{1 - t^{2}}} \Rightarrow \sqrt{a} f (a) = \int \frac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{2 t^{2}} d t

= \frac{1}{2} \int \frac{\sqrt{1 + t^{2}}}{t^{2}} d t - \frac{1}{2} \int \frac{\sqrt{1 - t^{2}}}{t^{2}} d t b u t b y p a r t s

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

= - \frac{\sqrt{1 + t^{2}}}{t} + l n (t + \sqrt{1 + t^{2}}) + c_{1} a l s o b y p s r t s

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

= - \frac{\sqrt{1 - t^{2}}}{t} - a r c s i n t + c_{2} \Rightarrow

f (a) = \frac{1}{2 \sqrt{a}} {- \frac{\sqrt{1 + t^{2}}}{t} + l n (t + \sqrt{1 + t^{2}}) + \frac{\sqrt{1 - t^{2}}}{t}

+ a r c s i n t} + C .