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Question-145609

Question Number 145609 by Khalmohmmad last updated on 06/Jul/21 Commented by imjagoll last updated on 06/Jul/21 $$\:\mathrm{let}\:\mathrm{g}\left(\mathrm{x}\right)=\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right) \\ $$$$\Rightarrow\left(\mathrm{g}\bullet\mathrm{f}\right)\left(\mathrm{x}\right)=\mathrm{x} \\ $$$$\Rightarrow\mathrm{f}\:'\left(\mathrm{x}\right).\mathrm{g}'\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{1} \\ $$$$\Rightarrow\mathrm{g}\:'\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\frac{\mathrm{1}}{\mathrm{f}\:'\left(\mathrm{x}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{8x}^{\mathrm{3}} +\mathrm{12x}^{\mathrm{2}}…

Question-145534

Question Number 145534 by wassim last updated on 05/Jul/21 Answered by Rasheed.Sindhi last updated on 06/Jul/21 $${C}-\mathrm{1}:{Finite}\:{sets}\:{containing}\: \\ $$$$\boldsymbol{{single}}\:\boldsymbol{{member}}: \\ $$$${Let}\:{x}\in\mathrm{M} \\ $$$$\therefore\:{x}^{\mathrm{2}} −\mathrm{3}\mid{x}\mid+\mathrm{4}={x} \\…

The-roots-of-the-equation-2x-2-px-q-0-are-2-and-2-Find-the-values-of-p-and-q-

Question Number 145443 by 7770 last updated on 04/Jul/21 $${The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} +{px}+{q}=\mathrm{0}\:\:{are}\:\mathrm{2}\alpha+\beta\:\:{and} \\ $$$$\alpha+\mathrm{2}\beta.\:{Find}\:{the}\:{values}\:{of}\:{p}\:{and}\:{q} \\ $$ Answered by Olaf_Thorendsen last updated on 05/Jul/21 $$\mathrm{S}\:=\:{x}_{\mathrm{1}}…

ln-cosx-dx-

Question Number 145412 by Jamshidbek last updated on 04/Jul/21 $$\:\:\:\:\int\mathrm{ln}\left(\mathrm{cosx}\right)\mathrm{dx}=? \\ $$ Answered by Olaf_Thorendsen last updated on 04/Jul/21 $$\mathrm{F}\left({x}\right)\:=\:\int\mathrm{ln}\left(\mathrm{cos}{x}\right)\:{dx} \\ $$$$\mathrm{F}\left({x}\right)\:=\:{x}\mathrm{ln}\left(\mathrm{cos}{x}\right)+\int{x}\mathrm{tan}{x}\:{dx} \\ $$$$\mathrm{F}\left({x}\right)\:=\:{x}\mathrm{ln}\left(\mathrm{cos}{x}\right)+ \\…

Let-f-0-1-R-be-a-differentiable-function-such-that-f-f-x-x-for-all-x-0-1-and-f-0-1-If-n-is-a-positive-integer-evaluate-the-following-integral-0-1-x-f-x-2n

Question Number 145318 by ZiYangLee last updated on 04/Jul/21 $$\mathrm{Let}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{differentiable}\:\mathrm{function} \\ $$$$\mathrm{such}\:\mathrm{that}\:{f}\left({f}\left({x}\right)\right)={x}\:\mathrm{for}\:\mathrm{all}\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{and} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{1}. \\ $$$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer},\:\mathrm{evaluate}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{integral}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({x}−{f}\left({x}\right)\right)^{\mathrm{2}{n}} \:{dx} \\ $$…