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

Question Number 40043 by abdo mathsup 649 cc last updated on 15/Jul/18

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

Answered by prof Abdo imad last updated on 22/Jul/18

c h a n g e m e n t \sqrt{x + 1} = t g i v e x + 1 = t^{2}

I = \int_{0}^{\infty} t e^{- (t^{2} - 1)} 2 t d t

= 2 e \int_{0}^{\infty} t^{2} e^{- t^{2}} d t a n d b y {p a r t s u}^{^{'}} = t e^{- t^{2}} v = t

I = 2 e {{[- \frac{1}{2} e^{- t^{2}} t]}_{0}^{+ \infty} + \int_{0}^{\infty} \frac{1}{2} e^{- t^{2}} d t}

= e \int_{0}^{\infty} e^{- t^{2}} d t = e \frac{\sqrt{π}}{2} \Rightarrow

I = \frac{e}{2} \sqrt{π} .