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Category: Integration

x-x-1-dx-

Question Number 129377 by pipin last updated on 15/Jan/21 $$\int\frac{\:\sqrt{\boldsymbol{\mathrm{x}}}}{\:\sqrt{\boldsymbol{\mathrm{x}}-\mathrm{1}}}\mathrm{dx}\:=\:… \\ $$ Answered by Ar Brandon last updated on 15/Jan/21 $$\mathcal{I}=\int\frac{\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}−\mathrm{1}}}\mathrm{dx}\:,\:\mathrm{x}=\mathrm{t}^{\mathrm{2}} \:\Rightarrow\mathrm{dx}=\mathrm{2tdt} \\ $$$$\:\:\:=\mathrm{2}\int\frac{\mathrm{t}^{\mathrm{2}} }{\:\sqrt{\mathrm{t}^{\mathrm{2}}…

let-f-a-dx-a-2-x-2-3-with-a-gt-0-1-calculate-f-a-2-calculste-also-g-a-dx-a-2-x-2-4-3-find-the-values-of-integrals-0-dx-x-2-1-3

Question Number 63782 by mathmax by abdo last updated on 09/Jul/19 $${let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({a}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){calculste}\:{also}\:{g}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{4}}…

cos-2-x-cos-x-1-sin-x-cos-x-dx-

Question Number 129274 by bramlexs22 last updated on 14/Jan/21 $$\:\int\:\frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$ Answered by liberty last updated on 14/Jan/21 $$\:\int\:\frac{\mathrm{cos}\:\mathrm{x}\left(\mathrm{cos}\:\mathrm{x}+\mathrm{1}\right)}{\mathrm{sin}\:\mathrm{x}+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\:\mathrm{dx}\:=\:\int\:\frac{\mathrm{cos}\:\mathrm{x}\left(\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{2cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left[\:\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right]}\:\mathrm{dx}= \\ $$$$\:\int\:\frac{\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left(\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)\left(\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\mathrm{dx}=…