find-the-value-of-0-1-e-x-1-e-2x-dx-

Question Number 38202 by prof Abdo imad last updated on 22/Jun/18

f i n d t h e v a l u e o f \int_{0}^{1} e^{- x} \sqrt{1 - e^{- 2 x}} d x

Commented by math khazana by abdo last updated on 23/Jun/18

c h a n g e m e n t e^{- x} = t g i v e x = - l n (t) a n d

I = - \int_{1}^{e^{- 1}} t \sqrt{1 - t^{2}} \frac{d t}{t}

= \int_{\frac{1}{e}}^{1} \sqrt{1 - t^{2}} d t a f t e r w e u s e t h e c h a n g . t = s i n θ

I = \int_{a r c s i n (e^{- 1})}^{\frac{π}{2}} c o s θ . c o s s θ d θ

= \int_{a r c s i n (e^{- 1})}^{\frac{π}{2}} \frac{1 + c o s (2 θ)}{2} d θ

= \frac{1}{2} (\frac{π}{2} - a r c s i n (e^{- 1})) + \frac{1}{4} {[s i n (2 θ)]}_{a r c s i n (e^{- 1})}^{\frac{π}{2}}

= \frac{π}{4} - \frac{1}{2} a r c s i n (e^{- 1}) - \frac{1}{4} s i n (2 a r c s i n (e^{- 1})) .