Category: Integration

Question Number 5421 by love math last updated on 14/May/16 $${\int }_0^2{xe}^{x^2}dx$$ Answered by nchejane last updated on 14/May/16 $$=\left[\frac{\mathrm{1}}{\mathrm{2}}{e}^{{x}^{\mathrm{2}}…

Question Number 136476 by Ar Brandon last updated on 22/Mar/21 $$\left(\mathrm{a}\right)\mathrm{Let}$$$$\mathrm{I}\left(\alpha \right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-{(\mathrm{x}-\frac{\alpha }{\mathrm{x}})}^2}\mathrm{dx}$$$$\mathrm{Show}\mathrm{that}\mathrm{it}\mathrm{is}\mathrm{legitimate}\mathrm{to}\mathrm{take}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{I}\left(\alpha \right)\mathrm{and}\mathrm{also}{\mathrm{I}}^{\prime }\left(\alpha \right)=$$$$0.\mathrm{Then}\mathrm{show}\mathrm{that}$$$$\mathrm{I}\left(\alpha \right)=\frac{\sqrt{\pi }}{2}.$$$$\left(\mathrm{b}\right)\:\mathrm{Use}\:\left(\mathrm{a}\right)\:\mathrm{to}\:\mathrm{prove}…

Question Number 5406 by 1771727373 last updated on 14/May/16 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{x}dx=?$$ Commented by FilupSmith last updated on 14/May/16 $${S}=\underset{{k}\rightarrow\infty} {\mathrm{lim}}\:\underset{−{k}} {\overset{{k}}…

Question Number 136445 by aupo14 last updated on 22/Mar/21 Answered by liberty last updated on 22/Mar/21 $$letf\left(x\right)={\int }_0^x\frac{t-3}{t^2+7}dt$$$$f{}^{\prime }\left(x\right)=\frac{x-3}{x^2+7}=0\Rightarrow x=3$$$${f}\:''\left({x}\right)\mid_{{x}=\mathrm{3}}…

Question Number 70909 by rajesh4661kumar@gmail.com last updated on 09/Oct/19 Commented by rajesh4661kumar@gmail.com last updated on 10/Oct/19 $$28$$ Commented by Abdo msup. last updated…