Menu Close

Category: Integration

8cos-3-x-8-sin-3-2x-dx-

Question Number 88863 by M±th+et£s last updated on 13/Apr/20 $$\int\frac{\mathrm{8}{cos}^{\mathrm{3}} \left({x}\right)}{\mathrm{8}+{sin}^{\mathrm{3}} \mathrm{2}{x}}{dx} \\ $$$$ \\ $$ Commented by MJS last updated on 13/Apr/20 $$\mathrm{I}\:\mathrm{tried}\:\mathrm{everything}\:\mathrm{I}\:\mathrm{know},\:\mathrm{seems}\:\mathrm{impossible} \\…

prove-that-0-n-x-dx-n-n-1-2-and-0-n-x-dx-n-n-1-2-when-is-floor-and-is-ceil-

Question Number 88852 by M±th+et£s last updated on 13/Apr/20 $${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{{n}} \lceil{x}\rceil{dx}=\:\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\:{and}\:\int_{\mathrm{0}} ^{{n}} \lfloor{x}\rfloor{dx}=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$$${when}\:\lfloor..\rfloor\:{is}\:{floor}\:{and}\:\lceil..\rceil\:{is}\:{ceil} \\ $$ Answered by mr W last…

Compute-the-area-of-a-loop-of-the-curve-asin-2-and-even-sketch-the-curve-please-

Question Number 23283 by ajfour last updated on 28/Oct/17 $${Compute}\:{the}\:{area}\:{of}\:{a}\:{loop}\:{of} \\ $$$${the}\:{curve}\:\boldsymbol{\rho}=\boldsymbol{{a}}\mathrm{sin}\:\mathrm{2}\boldsymbol{\theta}\:;\:{and}\:{even} \\ $$$${sketch}\:{the}\:{curve},\:{please}. \\ $$ Answered by mrW1 last updated on 28/Oct/17 $$\mathrm{a}\:\mathrm{small}\:\mathrm{loop}\:\mathrm{for}\:\theta\:\mathrm{from}\:\mathrm{0}\:\mathrm{to}\:\frac{\pi}{\mathrm{2}}: \\…

ln-x-1-dx-dy-

Question Number 88789 by M±th+et£s last updated on 12/Apr/20 $$\int\int{ln}\left({x}+\mathrm{1}\right)\:{dx}\:{dy} \\ $$ Commented by mr W last updated on 13/Apr/20 $$\int\:\mathrm{ln}\left(\mathrm{x}+\mathrm{1}\right)\:\mathrm{dx}\:=\:\int\:\mathrm{ln}\:\mathrm{u}\:\mathrm{du} \\ $$$$….. \\ $$$$=\:\left({x}+\mathrm{1}\right)\:\mathrm{ln}\left({x}+\mathrm{1}\right)−\left({x}+\mathrm{1}\right)+{c}\left({y}\right)…

Question-154318

Question Number 154318 by liberty last updated on 17/Sep/21 Answered by MJS_new last updated on 17/Sep/21 $$\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}\:=\sqrt{\mathrm{2}}\mathrm{sin}\:\frac{\mathrm{4}{x}+\pi}{\mathrm{4}}\:= \\ $$$$\:\:\:\:\:\left[\mathrm{sin}\:\theta\:=\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:\frac{\mathrm{2}\theta−\pi}{\mathrm{4}}\right] \\ $$$$=\sqrt{\mathrm{2}}\left(\mathrm{1}−\mathrm{2sin}^{\mathrm{2}} \:\frac{\mathrm{4}{x}−\pi}{\mathrm{8}}\right) \\ $$$$\int\sqrt{\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}}{dx}=…