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Category: Integration

Question-72346

Question Number 72346 by aliesam last updated on 27/Oct/19 Commented by mathmax by abdo last updated on 27/Oct/19 $${I}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}^{\mathrm{10}} +{x}^{\mathrm{6}} \:+{x}^{\mathrm{4}} \:+\mathrm{1}\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\:\:\:{by}\:{psrts}\:{u}^{'}…

Question-137876

Question Number 137876 by Bekzod Jumayev last updated on 07/Apr/21 Answered by MJS_new last updated on 07/Apr/21 $$\int\frac{{dx}}{\left({x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} }= \\ $$$$\:\:\:\:\:\left[{t}=\frac{{x}}{\left({x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} }\:\rightarrow\:{dx}=−\left({x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{4}/\mathrm{3}}…

nice-calculus-n-1-sin-nx-n-pi-2-x-2-Im-1-e-ix-Imln-1-cos-x-isin-x-Im-ln-1-cos-x-2-sin-2-x-itan-1

Question Number 137873 by mnjuly1970 last updated on 07/Apr/21 $$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:…….{nice}\:\:…………{calculus}……. \\ $$$$\:\:\:\:\boldsymbol{\phi}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({nx}\right)}{{n}}\:=\frac{\pi}{\mathrm{2}}−\frac{{x}}{\mathrm{2}} \\ $$$$\:\:\:\boldsymbol{\phi}=−{Im}\left(\mathrm{1}−{e}^{{ix}} \right)=−{Imln}\left\{\left(\mathrm{1}−{cos}\left({x}\right)−{isin}\left({x}\right)\right)\right\} \\ $$$$\:\:\:\:=−{Im}\left\{{ln}\left(\sqrt{\left(\mathrm{1}−{cos}\left({x}\right)\right)^{\mathrm{2}} +{sin}^{\mathrm{2}} \left({x}\right)}\:+{itan}^{−\mathrm{1}} \left(\frac{−{sin}\left({x}\right)}{\mathrm{1}−{cos}\left({x}\right)}\right)\right\}\right. \\…

Obain-an-equation-for-the-left-Reimen-Sum-the-right-Reimen-sum-Trapeziodal-rule-Newton-Raphson-s-Iteration-Hence-find-and-approximate-value-for-0-3-e-x-x-2-dx-

Question Number 72336 by Rio Michael last updated on 27/Oct/19 $${Obain}\:{an}\:{equation}\:{for}\: \\ $$$$\Rightarrow\:{the}\:{left}\:{Reimen}\:{Sum} \\ $$$$\Rightarrow\:{the}\:{right}\:{Reimen}\:{sum} \\ $$$$\Rightarrow\:{Trapeziodal}\:{rule} \\ $$$$\Rightarrow\:{Newton}\:{Raphson}'{s}\:{Iteration} \\ $$$$\:\:{Hence}\:{find}\:{and}\:{approximate}\:{value}\:{for}\:\int_{\mathrm{0}} ^{\mathrm{3}} \left({e}^{{x}} \:+\:{x}^{\mathrm{2}} \right){dx}…