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Category: Integration

dx-x-1-2-x-2-1-

Question Number 118145 by bemath last updated on 15/Oct/20 $$\int\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:? \\ $$ Commented by bobhans last updated on 15/Oct/20 $$\mathrm{Decomposition}\:\mathrm{fractional} \\ $$$$\:\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{2}}…

dx-sec-x-2-

Question Number 118094 by bemath last updated on 15/Oct/20 $$\int\:\frac{\mathrm{dx}}{\mathrm{sec}\:\mathrm{x}+\mathrm{2}}\:=? \\ $$ Answered by Dwaipayan Shikari last updated on 15/Oct/20 $$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{cosxdx}}{\mathrm{2}{cosx}+\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}\int\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{cosx}+\mathrm{1}}{dx} \\ $$$$=\frac{{x}}{\mathrm{2}}−\int\frac{\mathrm{1}}{\frac{\mathrm{2}−\mathrm{2}{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }+\mathrm{1}}.\frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}}…

suppose-f-1-3-1-1-such-that-1-3-f-x-dx-0-What-the-maximum-value-of-1-3-x-1-f-x-dx-

Question Number 118084 by bemath last updated on 15/Oct/20 $$\mathrm{suppose}\:\mathrm{f}:\:\left[\mathrm{1},\mathrm{3}\:\right]\rightarrow\:\left[−\mathrm{1},\mathrm{1}\:\right]\:\mathrm{such} \\ $$$$\mathrm{that}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{0}\:.\:\mathrm{What}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\:\mathrm{x}^{−\mathrm{1}} .\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$ Commented by john santu…

arc-cos-cos-x-1-2cos-x-dx-

Question Number 118081 by bemath last updated on 15/Oct/20 $$\:\int\:\mathrm{arc}\:\mathrm{cos}\:\left(\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{x}}\right)\:\mathrm{dx}\:=? \\ $$ Commented by MJS_new last updated on 15/Oct/20 $$\mathrm{seems}\:\mathrm{impossible}\:\mathrm{to}\:\mathrm{solve} \\ $$ Commented by MJS_new…