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Category: Integration

x-4-x-1-4-x-5-dx-

Question Number 128542 by bramlexs22 last updated on 08/Jan/21 $$\:\int\:\frac{\left(\mathrm{x}^{\mathrm{4}} −\mathrm{x}\right)^{\mathrm{1}/\mathrm{4}} }{\mathrm{x}^{\mathrm{5}} }\:\mathrm{dx}\:=? \\ $$ Answered by liberty last updated on 08/Jan/21 $$\:\gamma=\int\:\frac{\mathrm{x}\left(\mathrm{1}−\mathrm{x}^{−\mathrm{3}} \right)^{\mathrm{1}/\mathrm{4}} }{\mathrm{x}^{\mathrm{5}}…

If-f-x-lim-x-x-n-x-n-x-n-x-n-x-gt-1-then-xf-x-ln-x-1-x-2-1-x-2-dx-

Question Number 128540 by bramlexs22 last updated on 08/Jan/21 $$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{n}} −\mathrm{x}^{−\mathrm{n}} }{\mathrm{x}^{\mathrm{n}} +\mathrm{x}^{−\mathrm{n}} }\:,\mathrm{x}>\mathrm{1} \\ $$$$\mathrm{then}\:\int\:\frac{\mathrm{xf}\left(\mathrm{x}\right)\:\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\right)}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=? \\ $$ Commented by liberty last…

ln-1-xsin-2-x-sin-2-x-dx-

Question Number 62997 by aliesam last updated on 27/Jun/19 $$\int\frac{{ln}\left(\mathrm{1}+{xsin}^{\mathrm{2}} \left({x}\right)\right)}{{sin}^{\mathrm{2}} \left({x}\right)}\:{dx} \\ $$ Commented by mathmax by abdo last updated on 27/Jun/19 $${perhaps}\:{the}\:{Q}\:{is}\:{find}\:\int\:\frac{{ln}\left(\mathrm{1}+{tsin}^{\mathrm{2}} {x}\right)}{{sin}^{\mathrm{2}}…

Question-128529

Question Number 128529 by bramlexs22 last updated on 08/Jan/21 Commented by liberty last updated on 08/Jan/21 $$\mathrm{considering}\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{u}}{\mathrm{v}}\right)=\frac{\mathrm{u}'\mathrm{v}−\mathrm{uv}'}{\mathrm{v}^{\mathrm{2}} } \\ $$$$\:\mathrm{obvious}\:\mathrm{v}=\mathrm{2}+\mathrm{3cos}\:\mathrm{x}\:\mathrm{and}\:\mathrm{3}+\mathrm{2cos}\:\mathrm{x}=\mathrm{u}'\left(\mathrm{2}+\mathrm{3cos}\:\mathrm{x}\right)−\mathrm{u}\left(−\mathrm{3sin}\:\mathrm{x}\right) \\ $$$$\Rightarrow\mathrm{2cos}\:\mathrm{x}+\mathrm{3}=\mathrm{2u}'+\mathrm{3u}'\mathrm{cos}\:\mathrm{x}+\mathrm{3u}\:\mathrm{sin}\:\mathrm{x} \\ $$$$\mathrm{check}\:\mathrm{u}\:=\:\mathrm{sin}\:\mathrm{x}\:\Rightarrow\:\mathrm{u}'\left(\mathrm{2}+\mathrm{3cosx}\right)+\mathrm{3u}\:\mathrm{sin}\:\mathrm{x}=\: \\…

Question-62981

Question Number 62981 by hovea cw last updated on 27/Jun/19 Commented by mathmax by abdo last updated on 27/Jun/19 $$\left.\mathrm{2}\right)\:{let}\:{A}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{x}\mid{sin}\left(\mathrm{2}{x}\right)\mid}{\mathrm{3}+{sin}^{\mathrm{2}} {x}}\:{dx}\:{changement}\:{x}\:=\pi\:+{t}\:{give} \\ $$$${A}\:=\int_{−\pi}…