0-x-e-x-2-dx-evaluate-above-expression-

Question Number 8672 by swapnil last updated on 20/Oct/16

\underset{0}{\int^{\infty}} x . e^{- x^{2}} dx

evaluate above expression .

Answered by 123456 last updated on 21/Oct/16

u = - x^{2}

d u = - 2 x d x \Rightarrow x d x = - \frac{d u}{2}

x = 0, u = 0

x = ϵ, u = - ϵ^{2}

\underset{0}{\int^{\infty}} {x e}^{- x^{2}} d x = \lim_{ϵ \to \infty} \underset{0}{\int^{ϵ}} {x e}^{- x^{2}} d x

= \lim_{ϵ \to \infty} - \frac{1}{2} \underset{0}{\int^{- ϵ^{2}}} e^{u} d u

= \lim_{ϵ \to \infty} \frac{1}{2} \underset{- ϵ^{2}}{\int^{0}} e^{u} d u

= \lim_{ϵ \to \infty} \frac{1 - e^{- ϵ^{2}}}{2}

= \frac{1}{2}

Commented by swapnil last updated on 21/Oct/16

is ϵ > 0 ?

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