let-f-x-1-x-e-t-1-e-t-dt-with-x-lt-0-1-calculate-f-x-2-find-1-0-e-t-1-e-t-dt-

Question Number 40152 by maxmathsup by imad last updated on 16/Jul/18

l e t f (x) = \int_{- 1}^{x} \frac{e^{t}}{\sqrt{1 - e^{t}}} d t w i t h x < 0

1) c a l c u l a t e f (x)

2) f i n d \int_{- 1}^{0} \frac{e^{t}}{\sqrt{1 - e^{t}}} d t

Commented by maxmathsup by imad last updated on 16/Jul/18

c h a n g e m e n t e^{t} = u g i v e t = l n (u) \Rightarrow

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

f (x) = - 2 {\sqrt{1 - e^{x}} - \sqrt{1 - e^{- 1}}}

2) \int_{- 1}^{0} \frac{e^{t}}{\sqrt{1 - e^{t}}} d t = {l i m}_{x \to 0} f (x) = 2 \sqrt{1 - e^{- 1}}

Answered by tanmay.chaudhury50@gmail.com last updated on 16/Jul/18

\int_{- 1}^{x} \frac{e^{t}}{\sqrt{1 - e^{t}}} d t

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

= - 1 \times ∣ \frac{\sqrt{1 - e^{t}}}{\frac{1}{2}} ∣_{- 1}^{x}

= - 2 {\sqrt{1 - e^{x}} - \sqrt{1 - e^{- 1}}}

\int_{- 1}^{0} \frac{e^{t}}{\sqrt{1 - e^{t}}} d t

= - 2 {\sqrt{1 - e^{0}} - \sqrt{1 - e_{}^{- 1}}}

= 2 \sqrt{1 - e^{- 1}}}