Category: Integration

Question Number 26242 by abdo imad last updated on 22/Dec/17 $${prove}\:{that}\:\:\sum_{{k}=\mathrm{0}} ^{{k}={n}} \:\:{cos}^{\mathrm{2}} \left({kx}\right)=\:\frac{{n}+\mathrm{1}}{\mathrm{2}}\:\:+\:\frac{{sin}\left(\left({n}+\mathrm{1}\right){x}\right){cos}\left({nx}\right)}{\mathrm{2}\:{sinx}} \\ $$$${x}\:{from}\:{R}−\left\{\:{k}\pi.{k}\varepsilon{Z}\right\}{then}\:{find}\:{the}\:{value}\:{of}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}\left(\left({n}+\mathrm{1}\right){x}\right){cos}\left({nx}\right)}{{sinx}}{dx} \\ $$$$ \\ $$ Commented…

Question Number 157309 by cortano last updated on 22/Oct/21 $$\:\:\int\:\frac{{dx}}{\:\sqrt{{x}}+{x}\sqrt{{x}+\mathrm{1}}}\:=? \\ $$ Answered by MJS_new last updated on 22/Oct/21 $$\int\frac{{dx}}{\:\sqrt{{x}}+{x}\sqrt{{x}+\mathrm{1}}}= \\ $$$$=\int\frac{\sqrt{{x}+\mathrm{1}}}{{x}^{\mathrm{2}} +{x}−\mathrm{1}}{dx}−\int\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}−\mathrm{1}\right)\sqrt{{x}}} \\…

Question Number 91771 by  M±th+et+s last updated on 03/May/20 $$\int_{\mathrm{0}} ^{\infty} \frac{{sin}^{\mathrm{3}} \left({x}\right)}{{x}^{\mathrm{2}} }{dx} \\ $$ Commented by abdomathmax last updated on 03/May/20 $${let}\:{I}=\int_{\mathrm{0}} ^{\infty}…

Question Number 157268 by gsk2684 last updated on 21/Oct/21 $${prove}\:{that} \\ $$$$\int\frac{{x}^{\mathrm{2}} }{\left({x}\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} }{dx}=−\frac{{x}\mathrm{sec}\:{x}}{{x}\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}+\mathrm{tan}\:{x}+{c} \\ $$ Answered by Ar Brandon last updated on 18/Nov/21 $${I}=\int\frac{{x}^{\mathrm{2}}…

Question Number 157257 by john_santu last updated on 21/Oct/21 $$\int\:\frac{\mathrm{sin}\:^{\mathrm{6}} {x}+\mathrm{cos}\:^{\mathrm{5}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx} \\ $$ Answered by qaz last updated on 21/Oct/21 $$\int\frac{\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}+\mathrm{cos}\:^{\mathrm{5}}…