lim-x-1-1-x-e-t-2-dt-x-2-1-a-1-b-0-c-e-2-d-e-

Question Number 10829 by Saham last updated on 26/Feb/17

\lim_{x \to 1} \int_{1}^{x} \frac{e^{t^{2}} (dt)}{x^{2} - 1}

(a) 1 (b) 0 (c) e / 2 (d) e

Answered by bahmanfeshki last updated on 27/Feb/17

f (x) = \int_{1}^{x} e^{t^{2}} d t \Rightarrow f^{'} (x) = e^{x^{2}} \Rightarrow \lim_{x \to 1} \frac{1}{x^{2} - 1} \int_{1}^{x} e^{t^{2}} d t = \lim_{x \to 1} (\frac{\int_{1}^{x} e^{t^{2}} d t}{x - 1}) (\frac{1}{x + 1}) = f^{'} (1) \times \frac{1}{2} = \frac{e}{2}

Commented by FilupS last updated on 27/Feb/17

incorrect

Commented by bahmanfeshki last updated on 27/Feb/17

e d i t e d

Commented by Saham last updated on 27/Feb/17

God bless you sir .

Answered by FilupS last updated on 27/Feb/17

\lim_{x \to 1} \int_{1}^{x} \frac{e^{t^{2}}}{x^{2} - 1} d t

\lim_{x \to 1} \frac{1}{x^{2} - 1} \int_{1}^{x} e^{t^{2}} d t

\int e^{x^{2}} d x = \frac{1}{2} \sqrt{π} \erf (x)

\lim_{x \to 1} \frac{1}{x^{2} - 1} \times \frac{1}{2} \sqrt{π} (\erf (x) - \erf (1))

\frac{\sqrt{π}}{2} \lim_{x \to 1} \frac{\erf (x) - \erf (1)}{x^{2} - 1} = \frac{\sqrt{π}}{2} (\frac{0}{0})

L^{'} {Hopital}^{'} s Law

\frac{\sqrt{π}}{2} \lim_{x \to 1} \frac{\frac{d}{d x} (\erf (x) - \erf (1))}{\frac{d}{d x} (x^{2} - 1)}

\erf (1) = constant

\Rightarrow \erf (x) = \frac{2}{\sqrt{π}} \int e^{x^{2}} d x

\frac{\sqrt{π}}{2} \lim_{x \to 1} (\frac{\frac{2}{\sqrt{π}} e^{x^{2}} - 0}{2 x - 0})

\frac{\sqrt{π}}{2} \times \frac{2}{\sqrt{π}} \lim_{x \to 1} (\frac{e^{x^{2}}}{2 x})

\frac{e^{1}}{2}

∴ \lim_{x \to 1} \int_{1}^{x} \frac{e^{t^{2}}}{x^{2} - 1} d t = \frac{e}{2}

Commented by Saham last updated on 27/Feb/17

God bless you sir .

Commented by bahmanfeshki last updated on 27/Feb/17

m y a n s w e r i s e d i t e d \dots