2-2-sec-2-sec-1-x-x-x-2-1-

Question Number 176188 by gloriousman last updated on 14/Sep/22

\underset{\sqrt{2}}{\int^{2}} \frac{\sec^{2} (\sec^{- 1} x)}{x \sqrt{x^{2} - 1}}

Answered by BaliramKumar last updated on 14/Sep/22

\underset{\sqrt{2}}{\int^{2}} \frac{\sec^{2} (\sec^{- 1} x)}{x \sqrt{x^{2} - 1}} d x

l e t {s e c}^{- 1} (x) = y, w h e n x = \sqrt{2} t h e n y = \frac{π}{4}

\frac{d x}{x \sqrt{x^{2} - 1}} = d y, w h e n x = 2 t h e n y = \frac{π}{3}

\underset{\sqrt{2}}{\int^{2}} \frac{\sec^{2} (\sec^{- 1} x)}{x \sqrt{x^{2} - 1}} d x = \underset{\frac{π}{4}}{\int^{\frac{π}{3}}} {s e c}^{2} (y) \cdot d y

= {[t a n (y)]}_{\frac{π}{4}}^{\frac{π}{3}}

= t a n (\frac{π}{3}) - t a n (\frac{π}{4}) = \sqrt{3} - 1