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Category: Integration

Question-159597

Question Number 159597 by mathlove last updated on 19/Nov/21 Answered by mr W last updated on 19/Nov/21 $$\left(\frac{\mathrm{sin}^{\mathrm{2014}} \:{x}\:\mathrm{sin}\:\left(\mathrm{2014}{x}\right)}{\mathrm{2014}}\right)' \\ $$$$=\frac{\mathrm{2014}\:\mathrm{sin}^{\mathrm{2013}} \:{x}\:\mathrm{cos}\:{x}\:\mathrm{sin}\:\left(\mathrm{2014}{x}\right)+\mathrm{sin}\:^{\mathrm{2014}} {x}×\mathrm{2014}\:\mathrm{cos}\:\left(\mathrm{2014}{x}\right)}{\mathrm{2014}} \\ $$$$=\mathrm{sin}^{\mathrm{2013}}…

let-give-1-lt-a-lt-b-and-I-0-pi-a-b-du-x-cosu-dt-find-the-value-of-0-pi-ln-b-cost-a-cost-dt-

Question Number 28444 by abdo imad last updated on 25/Jan/18 $${let}\:{give}\:\:\mathrm{1}<{a}<{b}\:\:{and}\:{I}=\:\int_{\mathrm{0}} ^{\pi} \:\int_{{a}} ^{{b}} \:\:\frac{{du}}{{x}−{cosu}}\:{dt}\:\:{find}\:{the} \\ $$$${value}\:{of}\:\int_{\mathrm{0}} ^{\pi} \:\:{ln}\left(\frac{{b}−{cost}}{{a}−{cost}}\right){dt}\:. \\ $$ Terms of Service Privacy…

let-give-B-x-y-0-1-u-x-1-1-u-y-1-du-and-beta-function-and-x-0-u-x-1-e-u-du-x-gt-0-gamma-function-of-euler-1-prove-that-x-2-0-u-2x-1-e-u-2-du

Question Number 28442 by abdo imad last updated on 25/Jan/18 $${let}\:{give}\:{B}\left({x},{y}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{u}^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{y}−\mathrm{1}} {du}\:\:{and}\:\left({beta}\:{function}\right) \\ $$$${and}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{u}^{{x}−\mathrm{1}} \:{e}^{−{u}} \:{du}\:\:\:\:\:\left({x}>\mathrm{0}\right)\left({gamma}\:{function}\:{of}\:{euler}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:\:\Gamma\left({x}\right)=\:\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:{u}^{\mathrm{2}{x}−\mathrm{1}}…