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

$$\underset{x\to 1}{\lim }{\int }_1^{\mathrm{x}}\frac{{\mathrm{e}}^{{\mathrm{t}}^2}\left(\mathrm{dt}\right)}{{\mathrm{x}}^2-1}$$$$\left(\mathrm{a}\right)1\left(\mathrm{b}\right)0\left(\mathrm{c}\right)\mathrm{e}/2\left(\mathrm{d}\right)\mathrm{e}$$

Answered by bahmanfeshki last updated on 27/Feb/17

$$f\left(x\right)={\int }_1^x{e}^{t^2}dt\Rightarrow f^{\prime }\left(x\right)=e^{x^2}\Rightarrow \underset{x\to 1}{\lim }\frac{1}{x^2-1}{\int }_1^x{e}^{t^2}dt=\underset{x\to 1}{\lim }\left(\frac{{\int }_1^x{e}^{t^2}dt}{x-1}\right)\left(\frac{1}{x+1}\right)=f^{\prime }\left(1\right)\times \frac{1}{2}=\frac{e}{2}$$

Commented by FilupS last updated on 27/Feb/17

$$\mathrm{incorrect}$$

Commented by bahmanfeshki last updated on 27/Feb/17

$$edited$$

Commented by Saham last updated on 27/Feb/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Answered by FilupS last updated on 27/Feb/17

$$\underset{x\to 1}{\lim }{\int }_1^x\frac{e^{t^2}}{x^2-1}dt$$$$\underset{x\to 1}{\lim }\frac{1}{x^2-1}{\int }_1^x{e}^{t^2}dt$$$$\int e^{x^2}dx=\frac{1}{2}\sqrt{\pi }\mathrm{erf}\left(x\right)$$$$\underset{x\to 1}{\lim }\frac{1}{x^2-1}\times \frac{1}{2}\sqrt{\pi }(\mathrm{erf}\left(x\right)-\mathrm{erf}\left(1\right))$$$$\frac{\sqrt{\pi }}{2}\underset{x\to 1}{\lim }\frac{\mathrm{erf}\left(x\right)-\mathrm{erf}\left(1\right)}{x^2-1}=\frac{\sqrt{\pi }}{2}\left(\frac{0}{0}\right)$$$${\mathrm{L}}^{\prime }{\mathrm{Hopital}}^{\prime }\mathrm{s}\mathrm{Law}$$$$\frac{\sqrt{\pi }}{2}\underset{x\to 1}{\lim }\frac{\frac{d}{dx}(\mathrm{erf}\left(x\right)-\mathrm{erf}\left(1\right))}{\frac{d}{dx}(x^2-1)}$$$$\mathrm{erf}\left(1\right)=\mathrm{constant}$$$$\Rightarrow \mathrm{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}\int e^{x^2}dx$$$$\frac{\sqrt{\pi }}{2}\underset{x\to 1}{\lim }\left(\frac{\frac{2}{\sqrt{\pi }}{e}^{x^2}-0}{2x-0}\right)$$$$\frac{\sqrt{\pi }}{2}\times \frac{2}{\sqrt{\pi }}\underset{x\to 1}{\lim }\left(\frac{e^{x^2}}{2x}\right)$$$$\frac{e^1}{2}$$$$$$$$\therefore \underset{x\to 1}{\lim }{\int }_1^x\frac{e^{t^2}}{x^2-1}dt=\frac{e}{2}$$

Commented by Saham last updated on 27/Feb/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by bahmanfeshki last updated on 27/Feb/17

$$myanswerisedited...$$

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