Category: Integration

Question Number 200366 by ajfour last updated on 17/Nov/23 Commented by ajfour last updated on 17/Nov/23 $$Theredcirculararclengthisequal$$$$tobluearclengthofparabola.Find$$$$theequationofparabolaintheform$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}={ax}^{\mathrm{2}} +{c}. \…

Question Number 200254 by mnjuly1970 last updated on 16/Nov/23 $$$$$$calculate\dots$$$$\mathrm{\Omega }={\int }_{{\int }_0^{\frac{\pi }{2}}\ln \left(\tan \left(x\right)\right)dx}^{{\int }_0^{\mathrm{\infty }}\frac{{\sin }^2\left(x\right)}{x^2}dx}\ln \left(\sin \left(x\right)\right)dx=?$$ Answered…

Question Number 200299 by Calculusboy last updated on 16/Nov/23 Answered by witcher3 last updated on 17/Nov/23 $$\mathrm{x}\to \frac{1}{\mathrm{x}}$$$$\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 $${\ln }^2\left(\mathrm{x}\right)+1=\mathrm{y}\Rightarrow \mathrm{dy}=2\frac{\ln \left(\mathrm{x}\right)}{\mathrm{x}}$$$$=\int {\mathrm{y}}^2{\tan }^{-1}\left(\mathrm{y}\right)\mathrm{dy}$$$$=\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 $$s>0$$$$\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\sqrt{t}{\mathrm{e}}^{-st}dt=$$$$=−\frac{\mathrm{1}}{{s}}\left[\sqrt{{t}}\mathrm{e}^{−{st}} \right]_{\mathrm{0}}…