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Category: Integration

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

Question Number 85839 by sahnaz last updated on 25/Mar/20 $$\int\mathrm{x}×\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}\mathrm{dx} \\ $$ Commented by jagoll last updated on 25/Mar/20 $$\int\:\frac{\mathrm{x}\:\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{\mathrm{d}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{2}} }…

a-x-x-dx-

Question Number 20293 by tammi last updated on 25/Aug/17 $$\int\sqrt{\frac{{a}+{x}}{{x}}{dx}} \\ $$ Answered by $@ty@m last updated on 25/Aug/17 $$=\int\frac{{a}+{x}}{\:\sqrt{{x}\left({a}+{x}\right)}}{dx} \\ $$$$=\int\frac{{a}+{x}}{\:\sqrt{{ax}+{x}^{\mathrm{2}} }}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{x}+{a}+{a}}{\:\sqrt{{ax}+{x}^{\mathrm{2}}…

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

Question Number 20292 by tammi last updated on 25/Aug/17 $$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} +\sqrt{{x}−\mathrm{1}}} \\ $$ Answered by $@ty@m last updated on 25/Aug/17 $$=\int\frac{{dx}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{{x}−\mathrm{1}}} \\ $$$$=\int\frac{\sqrt{{x}+\mathrm{1}}−\sqrt{{x}−\mathrm{1}}}{\left({x}+\mathrm{1}\right)−\left({x}−\mathrm{1}\right)}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{{x}+\mathrm{1}}{dx}−\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{{x}−\mathrm{1}}{dx}…

1-x-1-x-dx-

Question Number 20291 by tammi last updated on 25/Aug/17 $$\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\:{dx}} \\ $$ Answered by $@ty@m last updated on 25/Aug/17 $$=\int\frac{\mathrm{1}−{x}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$$$=\int\frac{{dx}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{−\mathrm{2}{x}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}…

find-max-I-I-for-the-integral-I-z-z-1-cos-cos-cos-x-cos-cos-cos-x-dx-z-R-

Question Number 151360 by talminator2856791 last updated on 20/Aug/21 $$\: \\ $$$$\:\:\mathrm{find}\:\mathrm{max}\left(\Re\left(\mathrm{I}\right)+\Im\left(\mathrm{I}\right)\right)\:\mathrm{for}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\:\:\mathrm{I}\:=\:\int_{{z}} ^{\:{z}+\mathrm{1}} \:\mathrm{cos}\left(\mathrm{cos}\left(\mathrm{cos}\left({x}^{\mathrm{cos}\left(\mathrm{cos}\left(\mathrm{cos}\left({x}\right)\right)\right)} \right)\right)\right){dx} \\ $$$$\:\:{z}\:\in\:\mathbb{R} \\ $$$$\: \\ $$ Terms of…