Question Number 32039 by abdo imad last updated on 18/Mar/18 $${a}>−\mathrm{1}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{tan}^{\mathrm{2}} {t}}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+{atan}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{3}\right)\:\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{tan}^{\mathrm{2}}…
Question Number 163109 by mathmax by abdo last updated on 03/Jan/22 $$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cosx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{n}} }\mathrm{dx}\:\:\:\:\:\left(\mathrm{n}\:\mathrm{fromN}\:\mathrm{and}\:\mathrm{n}\geqslant\mathrm{1}\right) \\ $$ Answered by mathmax by abdo last updated…
Question Number 32034 by abdo imad last updated on 18/Mar/18 $${let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+{x}+…+{x}^{{n}} }\:\:{study}\:{the}\:{convergence}\:{of} \\ $$$$\Sigma\:{u}_{{n}} \:\:. \\ $$ Terms of Service Privacy Policy…
Question Number 97569 by M±th+et+s last updated on 08/Jun/20 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \sqrt{{tan}\left({x}\right)}\sqrt{\mathrm{1}−{tan}\left({x}\right)}\:{dx} \\ $$ Answered by MJS last updated on 08/Jun/20 $$\mathrm{use}\:\mathrm{thus}: \\ $$$${t}=\frac{\sqrt{\mathrm{1}−\mathrm{tan}\:{x}}}{\:\sqrt{\mathrm{tan}\:{x}}}\:\rightarrow\:{dx}=−\mathrm{2cos}^{\mathrm{2}} \:{x}\sqrt{\mathrm{tan}^{\mathrm{3}}…
Question Number 32031 by abdo imad last updated on 18/Mar/18 $${let}\:\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left({x}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} \left({xlnx}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{x}}…
Question Number 32029 by abdo imad last updated on 18/Mar/18 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\alpha{x}} {ln}\left({x}\right)\:{dx}\:\:{with}\:\:\alpha>\mathrm{0}\:. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 32026 by abdo imad last updated on 18/Mar/18 $${let}\:\alpha>\mathrm{0}\:{prove}\:{that}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\alpha}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\alpha−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\:. \\ $$ Commented by abdo imad last updated…
Question Number 97552 by bemath last updated on 08/Jun/20 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\sqrt{\frac{{x}}{\mathrm{1}−{x}}}\:\mathrm{ln}\left(\frac{{x}}{\mathrm{1}−{x}}\right)\:{dx}\:? \\ $$ Answered by john santu last updated on 08/Jun/20 Answered by abdomathmax…
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Question Number 163072 by abdullahhhhh last updated on 03/Jan/22 $$\int\frac{\boldsymbol{\mathrm{sinx}}+\boldsymbol{\mathrm{sin}}\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{sin}}\mathrm{5}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{sin}}\mathrm{7}\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{cosx}}+\boldsymbol{\mathrm{cos}}\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}\mathrm{5}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}\mathrm{7}\boldsymbol{\mathrm{x}}}\:\boldsymbol{\mathrm{dx}} \\ $$ Answered by tounghoungko last updated on 03/Jan/22 $$\:\frac{\mathrm{sin}\:\mathrm{7}{x}+\mathrm{sin}\:{x}+\mathrm{sin}\:\mathrm{5}{x}+\mathrm{sin}\:\mathrm{3}{x}}{\mathrm{cos}\:\mathrm{7}{x}+\mathrm{cos}\:{x}+\mathrm{cos}\:\mathrm{5}{x}+\mathrm{cos}\:\mathrm{3}{x}}\: \\ $$$$\:=\:\frac{\mathrm{2sin}\:\mathrm{4}{x}\:\mathrm{cos}\:\mathrm{3}{x}+\mathrm{2sin}\:\mathrm{4}{x}\:\mathrm{cos}\:{x}}{\mathrm{2cos}\:\mathrm{4}{x}\:\mathrm{cos}\:\mathrm{3}{x}+\mathrm{2cos}\:\mathrm{4}{x}\:\mathrm{cos}\:{x}} \\ $$$$=\:\frac{\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{cos}\:\mathrm{4}{x}} \\…