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Category: Integration

0-50pi-3-sinx-dx-

Question Number 136481 by BHOOPENDRA last updated on 22/Mar/21 $$\int_{\mathrm{0}} ^{\frac{\mathrm{50}\pi}{\mathrm{3}}} \mid{sinx}\mid{dx} \\ $$ Answered by MJS_new last updated on 22/Mar/21 $$\underset{\mathrm{0}} {\overset{\mathrm{50}\pi/\mathrm{3}} {\int}}\mid\mathrm{sin}\:{x}\mid{dx}=\mathrm{33}\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}}…

a-Let-I-0-e-x-x-2-dx-Show-that-it-is-legitimate-to-take-the-derivative-of-I-and-also-I-0-Then-show-that-

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

Question-136445

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