Question Number 142687 by iloveisrael last updated on 04/Jun/21 $$\:\:\:\:\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:{x}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}\:=? \\ $$ Answered by qaz last updated on 04/Jun/21 $$\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}\sqrt{\mathrm{1}+\mathrm{cos}\:\mathrm{x}}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int\frac{\mathrm{dx}}{\left(\mathrm{sin}\:\frac{\mathrm{x}}{\mathrm{2}}−\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int\frac{\mathrm{dx}}{\mathrm{2sin}\:\frac{\mathrm{x}}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}}−\mathrm{cos}\:^{\mathrm{2}} \frac{\mathrm{x}}{\mathrm{2}}}…
Question Number 11616 by Nayon last updated on 29/Mar/17 $${EvAluate}\:\int\left({x}^{\mathrm{2}} +\mathrm{9}\right)^{\mathrm{9}} {dx} \\ $$ Answered by Joel576 last updated on 29/Mar/17 $$\mathrm{Let}\:{u}\:=\:{x}^{\mathrm{2}} \:+\:\mathrm{9} \\ $$$$\frac{{du}}{{dx}}\:=\:\mathrm{2}{x}\:\:\Leftrightarrow\:\:{dx}\:=\:\frac{{du}}{\mathrm{2}{x}}\:\:…
Question Number 11609 by Nayon last updated on 29/Mar/17 $$ \\ $$$$ \\ $$$${Evaluate}\:\int{x}^{{x}} {dx} \\ $$ Commented by Nayon last updated on 29/Mar/17 $${pls}\:{ans}\:{mrw}\mathrm{1}…
Question Number 142667 by mnjuly1970 last updated on 03/Jun/21 Answered by mindispower last updated on 03/Jun/21 $${T}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {ln}\left(\mathrm{1}+{x}\right){dx}=\frac{{ln}\left(\mathrm{2}\right)}{{n}+\mathrm{1}}−\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}.\frac{{dx}}{\mathrm{1}+{x}} \\ $$$$\mathrm{0}\leqslant\int_{\mathrm{0}}…
Question Number 142656 by qaz last updated on 03/Jun/21 $$\int_{−\pi} ^{\pi} \frac{\mathrm{xsin}\:\mathrm{x}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}=? \\ $$ Answered by Ar Brandon last updated on 03/Jun/21 $$\xi\left(\mathrm{a}\right)=\int_{\mathrm{0}} ^{\pi}…
Question Number 142643 by ArielVyny last updated on 03/Jun/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{cos}^{\mathrm{2}} {t}}{{sint}}{dt} \\ $$ Answered by MJS_new last updated on 03/Jun/21 $$\int\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{sin}\:{t}}{dt}=\int\frac{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \:{t}}{\mathrm{sin}\:{t}}{dt}=\int\left(−\mathrm{sin}\:{t}\:+\mathrm{csc}\:{t}\right){dt}=…
Question Number 77103 by Boyka last updated on 03/Jan/20 Commented by turbo msup by abdo last updated on 03/Jan/20 $${let}\:{I}=\int\:\:\frac{\mathrm{2}}{\mathrm{2}+{sinx}}{dx}\:{changement} \\ $$$${tan}\left(\frac{{x}}{\mathrm{2}}\right)\:={t}\:{give}\: \\ $$$${I}=\int\:\:\frac{\mathrm{2}}{\mathrm{2}+\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }}×\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}}…
Question Number 142629 by qaz last updated on 03/Jun/21 $$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \centerdot\frac{\mathrm{2n}−\mathrm{1}}{\left(\mathrm{2n}\right)!}\centerdot\left(\frac{\pi}{\mathrm{2}}\right)^{\mathrm{2n}} \\ $$$$=\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{2n}−\mathrm{1}}{\left(\mathrm{2n}\right)!}\centerdot\left(−\left(\frac{\pi}{\mathrm{2}}\right)^{\mathrm{2}} \right)^{\mathrm{n}} \\ $$$$=\left(\mathrm{2xD}−\mathrm{1}\right)\mid_{\mathrm{x}=\pi/\mathrm{2}} \underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!}…
Question Number 77086 by Dah Solu Tion last updated on 03/Jan/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} {xtan}^{−\mathrm{1}} {xdx} \\ $$$$ \\ $$ Commented by Tony Lin last updated…
Question Number 77089 by Dah Solu Tion last updated on 03/Jan/20 $$\boldsymbol{{Cheap}}\:\downdownarrows \\ $$$$\int\sqrt{\frac{\boldsymbol{{x}}}{\:\sqrt{\frac{\boldsymbol{{x}}}{\:\sqrt{\frac{\boldsymbol{{x}}}{…}}}}}}\boldsymbol{{dx}} \\ $$$$ \\ $$$$ \\ $$ Commented by Tony Lin last…