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Question-41174

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Question Number 41174 by Tawa1 last updated on 02/Aug/18 Answered by MJS last updated on 03/Aug/18 $$\int\mathrm{e}^{{x}\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}} \left(\mathrm{cos}^{\mathrm{2}} \:{x}\:−\frac{\mathrm{sin}\:{x}}{{x}}−\frac{\mathrm{cos}\:{x}}{{x}^{\mathrm{2}} }\right){dx}= \\ $$$$\int\mathrm{e}^{{x}\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}} \mathrm{cos}^{\mathrm{2}} \:{x}\:{dx}−\int\frac{\mathrm{e}^{{x}\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}} \mathrm{sin}\:{x}}{{x}}{dx}−\int\frac{\mathrm{e}^{{x}\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}}…

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Question-106694

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Question Number 106694 by 175mohamed last updated on 06/Aug/20 Answered by bobhans last updated on 06/Aug/20 $$\mathrm{U}_{\mathrm{n}} \:=\:\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)}\:\Rightarrow\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\mathrm{40}} {\sum}}\:\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)} \\ $$$$\mathrm{decomposition}\:\mathrm{fraction}\: \\ $$$$\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)}\:=\:\frac{\mathrm{A}}{\mathrm{n}}\:+\:\frac{\mathrm{B}}{\mathrm{n}+\mathrm{1}}\:+\:\frac{\mathrm{C}}{\mathrm{n}+\mathrm{2}} \\…

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Question-172234

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Question Number 172234 by Sotoberry last updated on 24/Jun/22 Commented by cortano1 last updated on 24/Jun/22 $$\:{S}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \sqrt{\left(\frac{{dx}}{{d}\theta}\right)^{\mathrm{2}} +\left(\frac{{dy}}{{d}\theta}\right)^{\mathrm{2}} }\:{d}\theta \\ $$$$\:\begin{cases}{\frac{{dx}}{{d}\theta}\:=\:−\mathrm{3}{a}\mathrm{cos}\:^{\mathrm{2}} \theta\:\mathrm{sin}\:\theta\:}\\{\frac{{dy}}{{d}\theta}=\:\mathrm{2}{a}\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta}\end{cases} \\…

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