let-a-gt-2-and-f-a-1-a-1-a-x-2-dx-1-x-2-1-x-2-1-calculate-f-a-interms-of-a-2-calculate-f-a-

Question Number 49638 by maxmathsup by imad last updated on 08/Dec/18

l e t a > 2 a n d f (a) = \int_{- \frac{1}{a}}^{\frac{1}{a}} \frac{x^{2} d x}{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}}

1) c a l c u l a t e f (a) i n t e r m s o f a

2) c a l c u l a t e f^{^{'}} (a) .

Answered by Smail last updated on 08/Dec/18

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

= \int_{- 1 / a}^{1 / a} \frac{x^{2} (\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}})}{2 x^{2}} d x

= \frac{1}{2} \int_{- 1 / a}^{1 / a} (\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}) d x

\frac{1}{2} \int_{- 1 / a}^{1 / a} \sqrt{1 + x^{2}} d x - \frac{1}{2} \int_{- 1 / a}^{1 / a} \sqrt{1 - x^{2}} d x

x = s i n h (t) a n d x = s i n (u)

f (a) = \frac{1}{2} \int_{{s i n h}^{- 1} (- 1 / a)}^{{s i n h}^{- 1} (1 / a)} {c o s h}^{2} (t) d t - \frac{1}{2} \int_{{s i n}^{- 1} (- 1 / a)}^{{s i n}^{- 1} (1 / a)} {c o s}^{2} (u) d u

= \frac{1}{4} {[\frac{s i n h (2 t)}{2} + t]}_{{s i n h}^{- 1} (- 1 / a)}^{{s i n h}^{- 1} (1 / a)} - \frac{1}{4} {[\frac{c o s (2 u)}{2} + u]}_{{s i n}^{- 1} (- 1 / a)}^{{s i n}^{- 1} (1 / a)}

= \frac{1}{2} (\frac{1}{a} \sqrt{1 + {(\frac{1}{a})}^{2}} + {s i n h}^{- 1} (1 / a)) - \frac{1}{2} (\frac{1}{a} \sqrt{1 - \frac{1}{a^{2}}} + {s i n}^{- 1} (1 / a))

f (a) = \frac{1}{2} (\frac{\sqrt{a^{2} + 1}}{a^{2}} + {s i n h}^{- 1} (\frac{1}{a}) - \frac{\sqrt{a^{2} - 1}}{a^{2}} - {s i n}^{- 1} (1 / a))

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