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Question Number 151738 by tabata last updated on 22/Aug/21

$${\int }_0^{1}xd\left(e^{x^2}\right)$$$$$$$$howitsolve$$

Answered by Olaf_Thorendsen last updated on 22/Aug/21

$$\mathrm{I}={\int }_0^{1}xd\left(e^{x^2}\right)$$$$\mathrm{I}={\left[{xe}^{x^2}\right]}_0^1-{\int }_0^1{e}^{x^2}dx$$$$\mathrm{I}=e-\frac{\sqrt{\pi }}{2}{\left[\mathrm{erfi}\left(x\right)\right]}_0^1$$$$\mathrm{I}=e-\frac{\sqrt{\pi }}{2}\mathrm{erfi}\left(1\right)$$$$\mathrm{erfi}\left(z\right)=\frac{2}{\sqrt{\pi }}\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{z^{2n+1}}{(2n+1)n!}$$$$\mathrm{I}=e-\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{(2n+1)n!}$$

Commented by Olaf_Thorendsen last updated on 22/Aug/21

$$\mathrm{I}\approx e-1,462651746$$$$\mathrm{I}\approx 1,255630082$$

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