1-2-1-2-e-cos-1-x-1-x-2-dx-

Question Number 124654 by benjo_mathlover last updated on 05/Dec/20

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

Commented by mohammad17 last updated on 05/Dec/20

= - \int_{1 / \sqrt{2}}^{1 / 2} \frac{- 1}{\sqrt{1 - x^{2}}} \times e^{{c o s}^{- 1} (x)} d x

= - {[e^{{c o s}^{- 1} (x)}]}_{1 / \sqrt{2}}^{1 / 2} = - (e^{{c o s}^{- 1} (\frac{1}{2})} - e^{{c o s}^{- 1} (\frac{1}{\sqrt{2}})})

= e^{\frac{π}{4}} - e^{\frac{π}{3}}

(m o h a m m a d A l d o l a i m y)

Answered by mathmax by abdo last updated on 05/Dec/20

I = \int_{\frac{1}{\sqrt{2}}}^{\frac{1}{2}} \frac{e^{arcosx}}{\sqrt{1 - x^{2}}} dx cha 7 gement x = cost give

I = \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{e^{t}}{\sqrt{1 - \cos^{2} t}} (- sint) dt = - \int_{\frac{π}{4}}^{\frac{π}{3}} e^{t} dt = - {[e^{t}]}_{\frac{π}{4}}^{\frac{π}{3}}

= e^{\frac{π}{4}} - e^{\frac{π}{3}}

Answered by liberty last updated on 05/Dec/20

l e t z = \cos^{- 1} (x) \Rightarrow x = \cos z

d x = - \sin z d z

\int \frac{e^{z}}{\sqrt{1 - \cos z}} (- \sin z) d z = - \int e^{z} d z = - e^{z} + c

= - e^{\cos^{- 1} (x)} + c

\int_{\frac{1}{\sqrt{2}}}^{1 / 2} \frac{e^{\cos^{- 1} (x)}}{\sqrt{1 - x^{2}}} d x = - {[e^{\cos^{- 1} (x)}]}_{\frac{1}{\sqrt{2}}}^{\frac{1}{2}}

= - [e^{\frac{π}{3}} - e^{\frac{π}{4}}] = e^{\frac{π}{4}} - e^{\frac{π}{3}} .

Answered by Ar Brandon last updated on 05/Dec/20

Ω = \int_{1 / \sqrt{2}}^{1 / 2} \frac{e^{\cos^{- 1} (x)}}{\sqrt{1 - x^{2}}} dx

d (\cos^{- 1} (x)) = \frac{- 1}{\sqrt{1 - x^{2}}} dx

Ω = - \int_{1 / \sqrt{2}}^{1 / 2} e^{\cos^{- 1} (x)} d (\cos^{- 1} (x))

= - {[e^{\cos^{- 1} (x)}]}_{1 / \sqrt{2}}^{1 / 2} = e^{π / 4} - e^{π / 3}