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Category: Integration

let-f-x-0-1-dt-x-ch-t-1-find-a-explicite-form-of-f-x-2-calculate-0-1-dt-x-ch-t-2-3-find-the-value-of-0-1-dt-1-ch-t-and-0-1-dt-2-ch-t-4

Question Number 42487 by maxmathsup by imad last updated on 26/Aug/18 $${let}\:\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{dt}}{{x}\:+{ch}\left({t}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\left({x}+{ch}\left({t}\right)\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{ch}\left({t}\right)}\:{and}\:\:\int_{\mathrm{0}}…

Question-173523

Question Number 173523 by Physicien last updated on 13/Jul/22 Answered by Mathspace last updated on 13/Jul/22 $${I}=\int\:\:\frac{\mathrm{1}+\sqrt{{x}}−\sqrt{\mathrm{1}+{x}}}{\left(\mathrm{1}+\sqrt{{x}}\right)^{\mathrm{2}} −\mathrm{1}−{x}}{dx} \\ $$$$=\int\:\:\frac{\mathrm{1}+\sqrt{{x}}−\sqrt{\mathrm{1}+{x}}}{\mathrm{1}+\mathrm{2}\sqrt{{x}}+{x}−\mathrm{1}−{x}}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{\mathrm{1}+\sqrt{{x}}}{\:\sqrt{{x}}}{dx}−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\sqrt{\mathrm{1}+{x}}}{\:\sqrt{{x}}}{dx} \\ $$$${we}\:{have}\:\int\frac{\mathrm{1}+\sqrt{{x}}}{\:\sqrt{{x}}}{dx}=\int\frac{{dx}}{\:\sqrt{{x}}}\:+\int{dx} \\…