Category: Integration

Question Number 200366 by ajfour last updated on 17/Nov/23 Commented by ajfour last updated on 17/Nov/23 $${The}\:{red}\:{circular}\:{arc}\:{length}\:{is}\:{equal} \\ $$$$\:{to}\:{blue}\:{arclength}\:\:{of}\:{parabola}.\:{Find}\: \\ $$$${the}\:{equation}\:{of}\:{parabola}\:{in}\:{the}\:{form}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}={ax}^{\mathrm{2}} +{c}. \\…

Question Number 200254 by mnjuly1970 last updated on 16/Nov/23 $$ \\ $$$$\:\:\:\:\:\:{calculate}\:… \\ $$$$\:\:\Omega\:=\:\int_{\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\mathrm{ln}\left(\mathrm{tan}\left({x}\right)\right){dx}} ^{\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{sin}^{\mathrm{2}} \left({x}\right)}{{x}^{\mathrm{2}} }\:{dx}} \mathrm{ln}\left(\mathrm{sin}\left({x}\right)\right){dx}=? \\ $$ Answered…

Question Number 200299 by Calculusboy last updated on 16/Nov/23 Answered by witcher3 last updated on 17/Nov/23 $$\mathrm{x}\rightarrow\frac{\mathrm{1}}{\mathrm{x}} \\ $$$$\Omega=\int_{\infty} ^{\mathrm{0}} −\frac{\frac{\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{x}}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{4}} }.\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{x}^{\mathrm{2}}…

Question Number 200257 by Calculusboy last updated on 16/Nov/23 Answered by witcher3 last updated on 16/Nov/23 $$\mathrm{ln}^{\mathrm{2}} \left(\mathrm{x}\right)+\mathrm{1}=\mathrm{y}\Rightarrow\mathrm{dy}=\mathrm{2}\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}} \\ $$$$=\int\mathrm{y}^{\mathrm{2}} \mathrm{tan}^{−\mathrm{1}} \left(\mathrm{y}\right)\mathrm{dy} \\ $$$$=\frac{\mathrm{y}^{\mathrm{3}} }{\mathrm{3}}\mathrm{tan}^{−\mathrm{1}}…

Question Number 200155 by Calculusboy last updated on 15/Nov/23 Answered by Frix last updated on 15/Nov/23 $${s}>\mathrm{0} \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\sqrt{{t}}\mathrm{e}^{−{st}} {dt}= \\ $$$$=−\frac{\mathrm{1}}{{s}}\left[\sqrt{{t}}\mathrm{e}^{−{st}} \right]_{\mathrm{0}}…