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Question-69829

Question Number 69829 by TawaTawa last updated on 28/Sep/19 Commented by TawaTawa last updated on 28/Sep/19 $$\mathrm{The}\:\mathrm{question}\:\mathrm{says}\:\mathrm{do}\:\mathrm{not}\:\mathrm{use}\:\mathrm{Newton}'\mathrm{s}\:\mathrm{law}\:\mathrm{and}\:\mathrm{kinematic} \\ $$ Commented by TawaTawa last updated on…

0-1-1-6x-15x-2-20x-3-15x-4-6x-5-x-6-1-6-dx-pi-3-Or-0-1-1-kx-k-k-1-2-x-2-k-k-1-k-2-6-x-3-1-k-dx-pi-ksin-pi-k-

Question Number 135366 by Dwaipayan Shikari last updated on 12/Mar/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt[{\mathrm{6}}]{\mathrm{6}{x}−\mathrm{15}{x}^{\mathrm{2}} +\mathrm{20}{x}^{\mathrm{3}} −\mathrm{15}{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{5}} −{x}^{\mathrm{6}} }}{dx}=\frac{\pi}{\mathrm{3}} \\ $$$${Or} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt[{{k}}]{{kx}−\frac{{k}\left({k}−\mathrm{1}\right)}{\mathrm{2}}{x}^{\mathrm{2}} +\frac{{k}\left({k}−\mathrm{1}\right)\left({k}−\mathrm{2}\right)}{\mathrm{6}}{x}^{\mathrm{3}}…

find-dy-dx-at-the-point-0-3-when-2x-2-y-y-4xy-2-2x-3-

Question Number 69766 by Rio Michael last updated on 27/Sep/19 $${find}\:\:\frac{{dy}}{{dx}}\:\:{at}\:{the}\:{point}\:\:\left(\mathrm{0},\mathrm{3}\right)\:\:{when}\:\:\mathrm{2}{x}^{\mathrm{2}} {y}\:+\:{y}\:+\:\mathrm{4}{xy}^{\mathrm{2}} \:=\:\mathrm{2}{x}\:+\:\mathrm{3}\: \\ $$ Commented by kaivan.ahmadi last updated on 27/Sep/19 $${f}\left({x},{y}\right)=\mathrm{2}{x}^{\mathrm{2}} {y}+{y}+\mathrm{4}{xy}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{3}=\mathrm{0}…

Given-that-y-5x-2-3-show-that-when-x-2-6-5-d-2-y-dx-2-125-8-

Question Number 69765 by Rio Michael last updated on 27/Sep/19 $${Given}\:{that}\:\:{y}\:=\:\sqrt{\mathrm{5}{x}^{\mathrm{2}} \:+\:\mathrm{3}}\:,\:{show}\:{that}\:\:{when}\:{x}^{\mathrm{2}} \:=\:\frac{\mathrm{6}}{\mathrm{5}}\:,\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}\:} }\:=\:\frac{\mathrm{125}}{\mathrm{8}} \\ $$ Answered by MJS last updated on 27/Sep/19 $${y}=\sqrt{\mathrm{5}{x}^{\mathrm{2}}…

find-dy-dx-if-x-sin-2-t-and-y-tan-t-at-t-pi-4-

Question Number 69764 by Rio Michael last updated on 27/Sep/19 $${find}\:\:\:\frac{{dy}}{{dx}}\:\:{if}\:\:{x}\:=\:{sin}^{\mathrm{2}} {t}\:\:{and}\:\:{y}=\:{tan}\:{t}\:{at}\:\:{t}\:=\:\frac{\pi}{\mathrm{4}} \\ $$ Commented by kaivan.ahmadi last updated on 27/Sep/19 $$\frac{{dy}}{{dx}}=\frac{\frac{{dy}}{{dt}}}{\frac{{dx}}{{dt}}}=\frac{\mathrm{1}+{tan}^{\mathrm{2}} {t}}{{sin}\mathrm{2}{t}} \\ $$…

prove-by-mathematical-induction-that-for-all-positive-integers-n-r-1-n-r-r-1-n-3-n-1-n-2-

Question Number 69762 by Rio Michael last updated on 27/Sep/19 $${prove}\:{by}\:{mathematical}\:{induction},\:{that}\:{for}\:{all}\:{positive}\:{integers}\:{n}, \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left(\:{n}\:+\:\mathrm{2}\right) \\ $$ Commented by mathmax by abdo last updated on…