Question Number 217761 by mnjuly1970 last updated on 20/Mar/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:{prove}\:{that}\:: \\ $$$$ \\ $$$$ \\ $$$$\:\mathrm{I}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\pi{x}\right){sin}\left(\mathrm{2}\pi{x}\right){sin}\left(\mathrm{3}\pi{x}\right)}{{x}^{\mathrm{3}} }\:=\:\pi^{\mathrm{3}} \:\:\:\:\:\:\:\:\: \\ $$$$ \\…
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Question Number 217683 by MrGaster last updated on 18/Mar/25 $$\mathrm{Prove}:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\frac{\sqrt{{K}^{\mathrm{2}} +\mathrm{36}{K}'^{\mathrm{2}} }+\mathrm{6}{K}^{'} }{{K}^{\mathrm{2}} +\mathrm{36}{K}^{'\mathrm{2}} }\:}\frac{{dk}}{\:\sqrt{{k}}\left(\mathrm{1}−{k}^{\mathrm{2}} \right)^{\frac{\mathrm{2}}{\mathrm{3}}} }=\sqrt{\pi}\left(\sqrt{\mathrm{2}}−\sqrt{\frac{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}}}\right) \\ $$ Answered by MrGaster last…
Question Number 217626 by mnjuly1970 last updated on 17/Mar/25 $$ \\ $$$$\:\:\:\:\:\:\mathrm{lim}_{\:\lambda\rightarrow\mathrm{0}} \:\int_{\lambda} ^{\:\mathrm{2}\lambda} \:\frac{\:{e}^{\mathrm{2}{t}\:} }{{t}}\:{dt}\:=\:? \\ $$$$ \\ $$ Answered by maths2 last updated…
Question Number 217431 by peter frank last updated on 13/Mar/25 Answered by Frix last updated on 13/Mar/25 $$\mathrm{Simply}\:\mathrm{by}\:\mathrm{parts}: \\ $$$${u}'=\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:\rightarrow\:{u}=−\frac{\mathrm{1}}{{x}+\mathrm{1}} \\ $$$${v}={x}\mathrm{e}^{{x}} \:\rightarrow\:{v}'=\left({x}+\mathrm{1}\right)\mathrm{e}^{{x}} \\…
Question Number 217423 by MrGaster last updated on 13/Mar/25 Answered by MathematicalUser2357 last updated on 14/Mar/25 $$\mathrm{Triple}\:\mathrm{contour}\:\mathrm{integral} \\ $$$$\mathrm{Volume}\:\mathrm{integral} \\ $$ Terms of Service Privacy…
Question Number 217408 by Intesar last updated on 13/Mar/25 $${l}\int\mathrm{sin}\:\mathrm{7}{xdx} \\ $$ Answered by SdC355 last updated on 13/Mar/25 $$−\frac{\mathrm{1}}{\mathrm{7}}\mathrm{cos}\left(\mathrm{7}{x}\right)+{C} \\ $$$$\mathrm{because}. \\ $$$$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\:\mathrm{cos}\left({t}\right)=−\mathrm{sin}\left({t}\right) \\…
Question Number 217441 by mr W last updated on 13/Mar/25 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}^{\mathrm{3}} \:\left(\mathrm{1}−{x}\right)}{{x}^{\mathrm{3}} }\:{dx}=? \\ $$ Answered by MrGaster last updated on 14/Mar/25 $$\int_{\mathrm{0}}…
Question Number 217356 by SciMaths last updated on 11/Mar/25 Answered by profcedricjunior last updated on 11/Mar/25 $$\boldsymbol{{i}}=\int_{\mathrm{1}} ^{\mathrm{2}} \int_{\boldsymbol{{y}}} ^{\boldsymbol{{y}}^{\mathrm{2}} } \int_{\mathrm{0}} ^{\boldsymbol{{ln}}\left(\boldsymbol{{y}}+\boldsymbol{{z}}\right)} \boldsymbol{{e}}^{\boldsymbol{{x}}} \boldsymbol{{dxdydz}}…
Question Number 217289 by Frix last updated on 08/Mar/25 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}−{x}^{\mathrm{2}} }}\:{dx}=? \\ $$ Commented by Ghisom last updated on 11/Mar/25 $$\mathrm{we}\:\mathrm{can}\:\mathrm{use}\:\mathrm{partial}\:\mathrm{integration} \\…