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Category: Integration

du-u-2-1-u-

Question Number 83898 by sahnaz last updated on 07/Mar/20 $$\int\frac{\mathrm{du}}{\:\sqrt{\mathrm{u}^{\mathrm{2}} −\mathrm{1}\:}−\mathrm{u}} \\ $$ Commented by john santu last updated on 07/Mar/20 $$\mathrm{let}\:{u}\:=\:\mathrm{sec}\:\theta\:\Rightarrow\:\mathrm{du}\:=\:\mathrm{sec}\:\theta\:\mathrm{tan}\:\theta\:\mathrm{d}\theta \\ $$ Commented…

Defined-a-function-f-x-such-that-f-1-x-2f-x-nx-for-m-n-gt-1-the-value-of-1-m-2n-6f-m-x-dx-is-

Question Number 83899 by john santu last updated on 07/Mar/20 $$\mathrm{Defined}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{such}\: \\ $$$$\mathrm{that}\:\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right)+\mathrm{2f}\left(\mathrm{x}\right)=\:\mathrm{nx}\: \\ $$$$\mathrm{for}\:\mathrm{m}\:,\mathrm{n}\:>\:\mathrm{1}\:,\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\int\underset{\mathrm{1}} {\overset{\:\mathrm{m}} {\:}}\left(\mathrm{2n}+\mathrm{6f}\left(\frac{\mathrm{m}}{\mathrm{x}}\right)\right)\:\mathrm{dx}\:\mathrm{is}\:… \\ $$ Commented by john santu…

du-u-u-2-

Question Number 83893 by sahnaz last updated on 07/Mar/20 $$\int\frac{\mathrm{du}}{\mathrm{u}−\mathrm{u}^{\mathrm{2}} } \\ $$ Commented by abdomathmax last updated on 07/Mar/20 $$\int\:\:\frac{{du}}{{u}−{u}^{\mathrm{2}} }\:=\int\:\:\frac{{du}}{{u}\left(\mathrm{1}−{u}\right)}\:=\int\:\:\left(\frac{\mathrm{1}}{{u}}+\frac{\mathrm{1}}{\mathrm{1}−{u}}\right){du} \\ $$$$={ln}\mid\frac{{u}}{\mathrm{1}−{u}}\mid\:+{C} \\…

ln-x-ln-6x-x-2-dx-

Question Number 83842 by M±th+et£s last updated on 06/Mar/20 $$\int\frac{{ln}\left({x}\right)}{{ln}\left(\mathrm{6}{x}−{x}^{\mathrm{2}} \right)}{dx} \\ $$ Commented by Henri Boucatchou last updated on 06/Mar/20 $${Using}\:\:\:\frac{{lnA}}{{lnB}}=\frac{{A}}{{B}},\:\:\:\:\:\:\:\:\int\frac{{x}}{\mathrm{6}{x}−{x}^{\mathrm{2}} }{dx}=\int\left(−\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{2}{x}}{\mathrm{6}−{x}^{\mathrm{2}} }\right){dx}=−\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid\mathrm{6}−{x}^{\mathrm{2}} \mid\:+\:{Cte}…

If-the-function-of-f-is-continous-in-R-and-0-x-f-t-dt-x-1-t-2-f-t-dt-2x-2-4x-c-x-R-The-value-of-constant-c-is-

Question Number 83819 by jagoll last updated on 06/Mar/20 $$\mathrm{If}\:\mathrm{the}\:\mathrm{function}\:\mathrm{of}\:\mathrm{f}\:\mathrm{is}\:\mathrm{continous} \\ $$$$\mathrm{in}\:\mathbb{R}\:\mathrm{and}\:\int\underset{\mathrm{0}} {\overset{\:\mathrm{x}} {\:}}\:\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}\:=\:\int\underset{\mathrm{x}} {\overset{\:\mathrm{1}} {\:}}\mathrm{t}^{\mathrm{2}} \mathrm{f}\left(\mathrm{t}\right)\:\mathrm{dt}\:+\: \\ $$$$\mathrm{2x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{c}\:,\:\forall\mathrm{x}\in\mathbb{R}. \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{constant}\:\mathrm{c}\:\mathrm{is}\: \\ $$ Answered…