2-3-2-cos-sec-1-x-x-x-2-1-dx-2-2-sec-2-sec-1-x-x-x-2-1-dx-

Question Number 124644 by benjo_mathlover last updated on 05/Dec/20

\underset{2 / \sqrt{3}}{\int^{2}} \frac{\cos (\sec^{- 1} x)}{x \sqrt{x^{2} - 1}} d x

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

Answered by liberty last updated on 05/Dec/20

(1) \int \underset{\frac{2}{\sqrt{3}}}{\overset{2}{}} \frac{\cos (\sec^{- 1} x)}{x \sqrt{x^{2} - 1}} d x

p u t \sec^{- 1} x = j, x = \sec j a n d d x = \sec j \tan j d j

w h e r e u p p e r l i m i t j = \frac{π}{3} a n d l o w e r l i m i t j = \frac{π}{6}

\int {\overset{π / 3}{}}_{π / 6} \frac{\cos j}{\sec j \tan j} (\sec j \tan j) d j

I = {[\sin j]}_{\frac{π}{6}}^{\frac{π}{3}} = \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{1}{2} (\sqrt{3} - 1)

Answered by Dwaipayan Shikari last updated on 05/Dec/20

\int_{\sqrt{2}}^{2} \frac{{s e c}^{2} ({s e c}^{- 1} x)}{x \sqrt{x^{2} - 1}} d x {s e c}^{- 1} x = t \Rightarrow \frac{1}{x \sqrt{x^{2} - 1}} = \frac{d t}{d x}

= \int_{\frac{π}{4}}^{\frac{π}{3}} {s e c}^{2} (t) d t = {[t a n (t)]}_{\frac{π}{4}}^{\frac{π}{3}} = \sqrt{3} - 1