Category: Integration

Question Number 200606 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 20/Nov/23 $$\int \frac{\cos x{\sin }^{2}x}{\cos x+\sin x}dx\stackrel{t=x-\frac{\pi }{4}}{=}$$$$=\int (\frac{1}{4}+\frac{\cos t\sin t}{2}-\frac{{\sin }^{2}t}{2}-\frac{\tan t}{4})dt=$$$$=\frac{{t}}{\mathrm{4}}−\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{4}}−\frac{{t}+\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{4}}+\frac{\mathrm{ln}\:\mathrm{cos}\:{t}}{\mathrm{4}}=…

Question Number 200603 by Calculusboy last updated on 20/Nov/23 Answered by witcher3 last updated on 20/Nov/23 $${\mathrm{x}}^2={(\mathrm{n}-\mathrm{x})}^2-2\mathrm{n}(\mathrm{n}-\mathrm{x})+{\mathrm{n}}^2$$$$\int_{\mathrm{0}} ^{\mathrm{n}} \left(\mathrm{n}−\mathrm{x}\right)^{\mathrm{p}+\mathrm{2}} \mathrm{dx}−\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{n}}…

Question Number 200474 by Rupesh123 last updated on 19/Nov/23 Answered by witcher3 last updated on 19/Nov/23 $$\mathrm{erf}\left(\mathrm{x}\right)=\frac{2}{\sqrt{\pi }}{\int }_0^{\mathrm{x}}{\mathrm{e}}^{-{\mathrm{t}}^2}\mathrm{dt}$$$$\mathrm{ln}\left(\mathrm{x}+\mathrm{ln}\left(\mathrm{x}\right)\right)=\int_{\mathrm{0}} ^{\mathrm{5}} \mathrm{e}^{−\mathrm{t}^{\mathrm{2}}…

Question Number 200444 by Calculusboy last updated on 18/Nov/23 Answered by Frix last updated on 19/Nov/23 $$\int {\mathrm{e}}^{-\mathrm{i}{x}^2}dx\stackrel{t={\mathrm{e}}^{\mathrm{i}\frac{\pi }{4}}x}{=}$$$$=\frac{\sqrt{\mathrm{2}}\left(\mathrm{1}−\mathrm{i}\right)}{\mathrm{2}}\int\mathrm{e}^{−{t}^{\mathrm{2}} } {dt}=\frac{\sqrt{\mathrm{2}\pi}\left(\mathrm{1}−\mathrm{i}\right)}{\mathrm{4}}\int\frac{\mathrm{2e}^{{t}^{\mathrm{3}}…

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}. \…