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Category: Integration

given-f-x-f-x-pi-6-x-R-if-0-pi-6-f-x-dx-T-then-pi-7pi-3-f-x-pi-dx-

Question Number 103515 by bemath last updated on 15/Jul/20 $${given}\:{f}\left({x}\right)\:=\:{f}\left({x}+\frac{\pi}{\mathrm{6}}\right)\:\forall{x}\in\mathbb{R} \\ $$$${if}\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{6}} {\int}}{f}\left({x}\right){dx}\:=\:{T}\:{then}\:\underset{\pi} {\overset{\mathrm{7}\pi/\mathrm{3}} {\int}}{f}\left({x}+\pi\right) \\ $$$${dx}\:? \\ $$ Answered by bramlex last updated…

x-dx-cot-x-tan-x-2-a-x-16-x-sin-4x-32-cos-4x-128-c-b-x-16-x-sin-4x-32-cos-4x-128-c-c-x-16-xsin-4x-64-cos-4x-128-c-d-x-16-xcos-

Question Number 103512 by bemath last updated on 15/Jul/20 $$\int\:\frac{{x}\:{dx}}{\left(\mathrm{cot}\:{x}+\mathrm{tan}\:{x}\right)^{\mathrm{2}} }\:= \\ $$$$\left({a}\right)\:\frac{{x}}{\mathrm{16}}−\frac{{x}\:\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{32}}−\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$$$\left({b}\right)\:\frac{{x}}{\mathrm{16}}+\frac{{x}\:\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{32}}−\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$$$\left({c}\right)\:\frac{{x}}{\mathrm{16}}+\frac{{x}\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{64}}+\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$$$\left({d}\right)\frac{{x}}{\mathrm{16}}+\frac{{x}\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{32}}+\frac{\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$ Commented by bobhans last…

dx-x-x-1-4-1-a-9-x-1-4-1-18-x-1-4-1-9-c-b-9-x-1-4-1-18-x-1-4-1-9-c-c-9-x-1-4-1-18-x-1-4-1-9-c-d-9-x-

Question Number 103511 by bemath last updated on 15/Jul/20 $$\int\:\frac{{dx}}{\:\sqrt{{x}}\:\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)}\:=\_\_ \\ $$$$\left({a}\right)\:−\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}}{\mathrm{18}\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)^{\mathrm{9}} }\:+\:{c}\: \\ $$$$\left({b}\right)\:\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}}{\mathrm{18}\left(\sqrt[{\mathrm{4}}]{{x}}+\mathrm{1}\right)^{\mathrm{9}} }\:+{c} \\ $$$$\left({c}\right)\:−\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}\:−\mathrm{1}}{\mathrm{18}\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)^{\mathrm{9}} }\:+{c} \\ $$$$\left({d}\right)\:\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}+\mathrm{1}}{\mathrm{8}\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)^{\mathrm{9}} }\:+\:{c} \\ $$ Commented…

Question-37938

Question Number 37938 by Fawomath last updated on 19/Jun/18 Commented by abdo.msup.com last updated on 19/Jun/18 $$\left.\mathrm{1}\right)\:{let}\:{I}_{{p}} =\:\int\:{x}^{\mathrm{2}} \left({n}−{x}\right)^{{p}} \:{dx}\:{by}\:{parts} \\ $$$${I}_{{p}} =\:−\frac{\mathrm{1}}{{p}+\mathrm{1}}{x}^{\mathrm{2}} \left({n}−{x}\right)^{{p}+\mathrm{1}} \:\:+\frac{\mathrm{1}}{{p}+\mathrm{1}}\int\:\:\mathrm{2}{x}\:\:\left({n}−{x}\right)^{{p}+\mathrm{1}}…

f-N-R-g-N-R-f-n-0-2pi-x-n-sin-x-dx-g-n-0-2pi-x-n-cos-x-dx-f-n-1-f-n-g-n-1-g-n-

Question Number 37922 by gunawan last updated on 19/Jun/18 $${f}\::\:\mathbb{N}\:\rightarrow\:\mathbb{R} \\ $$$${g}\::\:\mathbb{N}\:\rightarrow\:\mathbb{R} \\ $$$${f}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} {x}^{{n}} \mathrm{sin}\:{x}\:{dx} \\ $$$${g}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} {x}^{{n}} \mathrm{cos}\:{x}\:{dx} \\ $$$$\frac{{f}\left({n}+\mathrm{1}\right)−{f}\left({n}\right)}{{g}\left({n}+\mathrm{1}\right)−{g}\left({n}\right)}=? \\…