Question Number 206541 by luciferit last updated on 18/Apr/24 Answered by lepuissantcedricjunior last updated on 18/Apr/24 $$\int\frac{\mathrm{3}\boldsymbol{{x}}+\mathrm{5}}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{10}\boldsymbol{{x}}+\mathrm{51}}}\boldsymbol{{dx}}=\boldsymbol{{k}} \\ $$$$\boldsymbol{{k}}=\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\mathrm{2}\boldsymbol{{x}}−\mathrm{5}+\frac{\mathrm{10}}{\mathrm{3}}+\mathrm{5}}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{10}\boldsymbol{{x}}+\mathrm{51}}}\boldsymbol{{dx}} \\ $$$$\boldsymbol{{k}}=\mathrm{3}\sqrt{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{10}\boldsymbol{{x}}+\mathrm{51}}+\frac{\mathrm{25}}{\mathrm{2}}\int\frac{\boldsymbol{{dx}}}{\:\sqrt{\left(\boldsymbol{{x}}−\mathrm{5}\right)^{\mathrm{2}} +\mathrm{26}}}…
Question Number 206543 by luciferit last updated on 18/Apr/24 Answered by lepuissantcedricjunior last updated on 18/Apr/24 $$\int\frac{\mathrm{3}+\mathrm{2}\sqrt{\boldsymbol{{x}}}}{\mathrm{4}+\sqrt{\boldsymbol{{x}}}}\boldsymbol{{dx}}=\int\frac{\mathrm{2}\left(\mathrm{4}+\sqrt{{x}}\right)−\mathrm{5}}{\mathrm{4}+\sqrt{\boldsymbol{{x}}}}\boldsymbol{{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\boldsymbol{{x}}−\left\{\mathrm{5}\int\frac{\boldsymbol{{dx}}}{\mathrm{4}+\sqrt{\boldsymbol{{x}}}}\:\:\:\:\boldsymbol{{x}}=\boldsymbol{{t}}^{\mathrm{2}} \Leftrightarrow\boldsymbol{{dx}}=\mathrm{2}\boldsymbol{{tdt}}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\boldsymbol{{x}}−\left\{\mathrm{10}\int\left(\mathrm{1}−\frac{\mathrm{4}}{\mathrm{4}+\boldsymbol{{t}}}\right)\boldsymbol{{dt}}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\boldsymbol{{x}}−\mathrm{10}\sqrt{\boldsymbol{{x}}}+\mathrm{40}\boldsymbol{{ln}}\left(\mathrm{4}+\sqrt{\boldsymbol{{x}}}\right)+\boldsymbol{{c}} \\…
Question Number 206542 by luciferit last updated on 18/Apr/24 Answered by lepuissantcedricjunior last updated on 18/Apr/24 $$\int\left(\mathrm{3}\boldsymbol{{x}}+\mathrm{5}\right)\boldsymbol{{arctan}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}=\boldsymbol{{k}} \\ $$$$\begin{cases}{\boldsymbol{{u}}=\boldsymbol{{arctan}}\left(\boldsymbol{{x}}\right)}\\{\boldsymbol{{v}}'=\left(\mathrm{3}\boldsymbol{{x}}+\mathrm{5}\right)}\end{cases}=>\begin{cases}{\boldsymbol{{u}}'=\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }}\\{\boldsymbol{{v}}=\frac{\mathrm{3}}{\mathrm{2}}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\boldsymbol{{x}}}\end{cases} \\ $$$$\boldsymbol{{k}}=\left(\frac{\mathrm{3}}{\mathrm{2}}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\boldsymbol{{x}}\right)\boldsymbol{{arctan}}\left(\boldsymbol{{x}}\right)−\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}+\frac{\mathrm{10}}{\mathrm{3}}\boldsymbol{{x}}−\mathrm{1}}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}}…
Question Number 206536 by cortano21 last updated on 18/Apr/24 $$\:\:\:\:\underline{\underbrace{\lessdot}\cancel{} }\underbrace{\nsupseteqq\spadesuit\left[}\mathrm{2}^{{x}} \:\mathrm{sin}\:\left(\sqrt{\mathrm{4}−\mathrm{2}^{{x}+\mathrm{2}} }\:\right)\right. \\ $$ Commented by Frix last updated on 18/Apr/24 $$\mathrm{Simply}\:\mathrm{use}\:{t}=\sqrt{\mathrm{1}−\mathrm{2}^{{x}} } \\…
Question Number 206568 by universe last updated on 18/Apr/24 Answered by Berbere last updated on 19/Apr/24 $$\frac{{x}}{{n}}={y}\:\:{A}\left({n}\right)=\int_{\mathrm{0}} ^{{n}} \left(\frac{\mathrm{2}{nx}}{{x}^{\mathrm{2}} +{n}^{\mathrm{2}} }\right)^{{n}} {dx}={n}\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\mathrm{2}{y}}{\mathrm{1}+{y}^{\mathrm{2}} }\right)^{{n}}…
Question Number 206537 by necx122 last updated on 18/Apr/24 $$\int\frac{{x}^{\mathrm{4}} −\mathrm{1}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}}{dx} \\ $$ Answered by Berbere last updated on 18/Apr/24 $$=\int\frac{{x}\left(\mathrm{4}{x}^{\mathrm{3}} +\mathrm{2}{x}\right)−\mathrm{2}{x}^{\mathrm{4}}…
Question Number 206489 by Simurdiera last updated on 16/Apr/24 $${Resuelve}\:{la}\:{siguiente}\:{integral} \\ $$$$\int\:\frac{\mathrm{sin}\:\left({t}\right)}{\:\sqrt[{\mathrm{4}}]{\mathrm{sin}^{\mathrm{7}} \left({t}\right)\centerdot\mathrm{cos}^{\mathrm{5}} \left({t}\right)}}\:{dt} \\ $$ Answered by Frix last updated on 16/Apr/24 $${t}=\mathrm{tan}\:{x} \\…
Question Number 206490 by Simurdiera last updated on 16/Apr/24 $${Resuelve}\:{la}\:{siguiente}\:{integral} \\ $$$$\int\:\frac{\mathrm{cos}\:\left({t}\right)}{\:\sqrt[{\mathrm{4}}]{\mathrm{sin}^{\mathrm{7}} \left({t}\right)\centerdot\mathrm{cos}^{\mathrm{5}} \left({t}\right)}}\:{dt} \\ $$ Answered by Frix last updated on 16/Apr/24 $${t}=\mathrm{tan}\:{x} \\…
Question Number 206338 by mnjuly1970 last updated on 12/Apr/24 Answered by Sorena last updated on 12/Apr/24 $$\left[{x}\right]−\left[{x}^{\mathrm{2}} \right]\geqslant\mathrm{0}\:\rightarrow\:\left[{x}\right]\geqslant\left[{x}^{\mathrm{2}} \right]\:\rightarrow\:{x}\in\left[\mathrm{0},\sqrt{\mathrm{2}}\right) \\ $$ Terms of Service Privacy…
Question Number 206248 by MetaLahor1999 last updated on 10/Apr/24 $$\int\frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}−{t}\right)\left(\mathrm{2}−{t}\right)}}{dt}=…? \\ $$ Answered by TonyCWX08 last updated on 10/Apr/24 $$\int\frac{\mathrm{1}}{\:\sqrt{{t}^{\mathrm{2}} −\mathrm{3}{t}+\mathrm{2}}}{dt} \\ $$$$=\int\frac{\mathrm{1}}{\:\sqrt{\left({t}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}}{dt} \\…