Question Number 219305 by Nicholas666 last updated on 22/Apr/25 $$ \\ $$$$\:\:\:\:\:\:{Prove}; \\ $$$$\:\:\:{I}_{\mathrm{0}} \left({x}\right)\:=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:{e}^{\:{x}\:{cox}\left(\theta\right)} \:{d}\theta\:; \\ $$$$\:\:\:{x}^{\mathrm{2}} {I}_{\mathrm{0}} ^{''} \left({x}\right)\:+\:{xI}'_{\mathrm{0}} \left({x}\right)\:−\:{x}^{\mathrm{2}} {I}_{\mathrm{0}}…
Question Number 219268 by MrGaster last updated on 21/Apr/25 $${f}\left({x},{y}\right)=\mathrm{ln}\int_{\mathrm{0}} ^{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } {e}^{{t}^{\mathrm{2}} } {dt},{f}\left({x}\right)\left(\mathrm{1},\mathrm{2}\right)=? \\ $$ Answered by zetamaths last updated on 21/Apr/25…
Question Number 219193 by fantastic last updated on 20/Apr/25 $$\int\sqrt{\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}}\:.\frac{\mathrm{1}}{{x}+\mathrm{3}}\:{dx}=? \\ $$ Answered by aleks041103 last updated on 20/Apr/25 $$\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}={u}=\mathrm{1}−\frac{\mathrm{1}}{{x}+\mathrm{2}} \\ $$$$\Rightarrow{x}=\frac{\mathrm{1}}{\mathrm{1}−{u}}−\mathrm{2}=\frac{\mathrm{2}{u}−\mathrm{1}}{\mathrm{1}−{u}} \\ $$$$\Rightarrow{dx}=\frac{\mathrm{2}\left(\mathrm{1}−{u}\right)−\left(−\mathrm{1}\right)\left(\mathrm{2}{u}−\mathrm{1}\right)}{\left(\mathrm{1}−{u}\right)^{\mathrm{2}} }{du}=…
Question Number 219236 by Nicholas666 last updated on 20/Apr/25 $$ \\ $$$$\:\:\:\:{f}\left({t}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\int_{\:{c}−{i}\infty} ^{\:{c}+{i}\infty} \:\frac{{e}^{{st}} }{{s}^{{k}} \:}\:\:{ds}\:\:\:,\:\:{k}\:\in\mathbb{C} \\ $$$$\: \\ $$ Commented by Nicholas666 last updated…
Question Number 219234 by Nicholas666 last updated on 22/Apr/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:{f}\left({t}\right)=\int_{\mathrm{0}\:} ^{\:{t}} \:\frac{\zeta\left(\mathrm{1}/\mathrm{2}\:\:+\:\:{i}\tau\right)}{\:\sqrt{{t}\:−\:\tau\:\:+\:\mathrm{1}}}\:{d}\tau \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 219060 by MrGaster last updated on 19/Apr/25 $$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}^{{m}} }{{x}^{{n}} }{dx},{n}\in\mathbb{N},{m}\in\mathbb{N},{n}\leqslant{m} \\ $$ Answered by Nicholas666 last updated on 19/Apr/25 $$\frac{\pi}{\mathrm{2}} \\…
Question Number 219077 by zetamaths last updated on 19/Apr/25 $$\int_{\mathrm{0}} ^{+\infty} \left(\frac{{sin}\left({n}\right)}{{n}}\right)^{{m}} {dn}=\pi\centerdot\frac{{m}}{\mathrm{2}^{{m}} }\centerdot\underset{\phi=\mathrm{0}} {\overset{{m}/\mathrm{2}} {\sum}}\left(−\mathrm{1}\right)^{\emptyset} \centerdot\frac{\left({n}−\mathrm{2}\phi\right)^{{m}−\mathrm{1}} }{\left({m}−\phi\right)!\centerdot\phi!}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Proof}\:{this}\:{formula} \\ $$ Terms of Service Privacy Policy…
Question Number 219078 by zetamaths last updated on 19/Apr/25 $$\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{m}}\right).\underset{{k}=\mathrm{1}} {\overset{{m}} {\sum}}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \centerdot{k}\centerdot\frac{\left({m}!\right)^{\mathrm{2}} }{\left({m}−{k}\right)!\left({m}+{k}\right)!}=\frac{\mathrm{1}}{\mathrm{4}}\:\:\:\:\:\:{Proof}\:{this}\:{formula} \\ $$ Answered by MrGaster last updated on 19/Apr/25 $$\mathrm{Let}\:{S}\left({m}\right)\underset{{k}=\mathrm{1}} {\overset{{m}}…
Question Number 218907 by malwan last updated on 17/Apr/25 $$\:_{\mathrm{0}} \int^{\:\mathrm{45}} {arctan}\left(\frac{\mathrm{1}+{tan}\:{x}}{\:\sqrt{\mathrm{2}}}\right){dx}\:=\:? \\ $$ Commented by mr W last updated on 17/Apr/25 $${you}\:{should}\:{make}\:{clear}\:{what}\:{you} \\ $$$${mean}\:{with}\:\int_{\mathrm{0}}…
Question Number 218896 by Nicholas666 last updated on 17/Apr/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{prove}}; \\ $$$$\:\mid\int\int\int_{\left[\mathrm{0},\infty\right]^{\mathrm{3}} } \boldsymbol{{f}}\frac{\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{x}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{y}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{z}}\right)}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} }\mid\leqslant\boldsymbol{{C}}\left(\int\int\int_{\mathbb{R}_{+} ^{\mathrm{3}} } \mid\boldsymbol{{f}}\mid\left(\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}}…