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Author: Tinku Tara

Question-212762

Question Number 212762 by universe last updated on 23/Oct/24 Commented by MrGaster last updated on 23/Oct/24 $$\mathrm{Rewrite}\:\mathrm{the}\:\mathrm{integrand}\:\mathrm{function}\:\mathrm{as}: \\ $$$$\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\ldots\left({x}−{n}\right)=\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}{x}\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\ldots\left({x}−{n}\right) \\ $$$$\mathrm{Computational}\:\mathrm{integral}: \\ $$$$\int_{\mathrm{0}} ^{{n}}…

Question-212784

Question Number 212784 by Spillover last updated on 23/Oct/24 Answered by A5T last updated on 24/Oct/24 $${d}^{\mathrm{2}} =\left({r}−{a}\right)^{\mathrm{2}} +{r}^{\mathrm{2}} −\mathrm{2}{r}\left({r}−{a}\right){cos}\theta \\ $$$${c}^{\mathrm{2}} =\left({r}−{a}\right)^{\mathrm{2}} +{r}^{\mathrm{2}} +\mathrm{2}{r}\left({r}−{a}\right){cos}\theta…

question-212462-prove-that-0-dx-x-4-25x-2-160-0-dx-x-4-95x-2-2560-my-attempt-but-is-it-a-proof-I-b-a-0-dx-x-4-ax-2-b-2-pi-b-1-4-agm-1

Question Number 212777 by Ghisom last updated on 23/Oct/24 $$\mathrm{question}\:\mathrm{212462} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{dx}}{\:\sqrt{{x}^{\mathrm{4}} +\mathrm{25}{x}^{\mathrm{2}} +\mathrm{160}}}=\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{dx}}{\:\sqrt{{x}^{\mathrm{4}} −\mathrm{95}{x}^{\mathrm{2}} +\mathrm{2560}}} \\ $$$$ \\…

Question-212720

Question Number 212720 by Spillover last updated on 22/Oct/24 Answered by A5T last updated on 22/Oct/24 $${AB}=\sqrt{\mathrm{2}\left(\mathrm{3}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} }=\mathrm{6};\:{Let}\:\angle{OAB}=\theta \\ $$$$\frac{{sin}\theta}{{r}}=\frac{{sin}\left(\mathrm{180}−\mathrm{2}\theta\right)}{{AB}}\Rightarrow{cos}\theta=\frac{\mathrm{3}}{{r}}\Rightarrow{sin}\mathrm{2}\theta=\frac{\mathrm{6}\sqrt{{r}^{\mathrm{2}} −\mathrm{9}}}{{r}^{\mathrm{2}} } \\ $$$$\left(\mathrm{4}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} ={r}^{\mathrm{2}}…