Question Number 139402 by mathmax by abdo last updated on 26/Apr/21 $$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{log}^{\mathrm{2}} \mathrm{x}}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$ Answered by mathmax by abdo last updated…
Question Number 8296 by tawakalitu last updated on 06/Oct/16 $$\mathrm{Give}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{representation}\:\mathrm{of}\:\:\mathrm{2333}! \\ $$ Commented by 123456 last updated on 07/Oct/16 $${n}!=\underset{\mathrm{0}} {\overset{+\infty} {\int}}{t}^{{n}} {e}^{−{t}} {dt} \\…
Question Number 73832 by FCB last updated on 16/Nov/19 Commented by FCB last updated on 16/Nov/19 $$\boldsymbol{\mathrm{solve}} \\ $$ Commented by MJS last updated on…
Question Number 8281 by tawakalitu last updated on 06/Oct/16 $$\int\frac{\mathrm{6}\:\mathrm{sinx}\:\mathrm{cosx}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\:\mathrm{dx} \\ $$ Answered by Yozzias last updated on 06/Oct/16 $$\frac{\mathrm{sinx}}{\mathrm{sinx}+\mathrm{cosx}}=\mathrm{1}−\frac{\mathrm{cosx}}{\mathrm{sinx}+\mathrm{cosx}} \\ $$$$\therefore\:\mathrm{I}=\int\frac{\mathrm{sinxcosx}}{\mathrm{sinx}+\mathrm{cosx}}\mathrm{dx}=\int\left(\mathrm{1}−\frac{\mathrm{cosx}}{\mathrm{sinx}+\mathrm{cosx}}\right)\mathrm{cosxdx} \\ $$$$=\int\left(\mathrm{cosx}−\frac{\mathrm{cos}^{\mathrm{2}} \mathrm{x}}{\mathrm{sinx}+\mathrm{cosx}}\right)\mathrm{dx}…
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Question Number 73751 by ~blr237~ last updated on 15/Nov/19 $${Find}\:\:{out}\:{the}\:{value}\:{of}\:\:\: \\ $$$$\:\:{J}=\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{2}{e}^{−\mathrm{2}{xy}} −{e}^{−{xy}} \right){dxdy}\: \\ $$ Commented by mathmax by abdo…
Question Number 139283 by qaz last updated on 25/Apr/21 $$\int_{\mathrm{0}} ^{\infty} \frac{{ln}^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{8}{a}}\left(\pi^{\mathrm{2}} +\mathrm{4}{ln}^{\mathrm{2}} {a}\right)\:\:\:\:\:\:\:\:\:,{a}>\mathrm{0} \\ $$ Answered by Ar Brandon last updated…
Question Number 139282 by qaz last updated on 25/Apr/21 $$\int_{\mathrm{0}} ^{\infty} \frac{{ln}\:{x}}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{2}{a}}\centerdot{ln}\:{a}\:\:\:\:\:\:\:\:\:\:,{a}>\mathrm{0}\: \\ $$ Answered by Ar Brandon last updated on 25/Apr/21 $$\Omega=\int_{\mathrm{0}}…
Question Number 139272 by bobhans last updated on 25/Apr/21 $$ \\ $$What is the area bounded by the parabola y^2=4ax and x^2 = 4ay? Answered…
Question Number 139259 by bobhans last updated on 25/Apr/21 $$\:\int\:\frac{\mathrm{cos}\:\mathrm{x}+\sqrt[{\mathrm{3}}]{\mathrm{7}}}{\mathrm{sin}\:\mathrm{x}+\sqrt{\mathrm{6}}}\:\mathrm{dx}\:=? \\ $$ Answered by Dwaipayan Shikari last updated on 25/Apr/21 $$\int\frac{{cosx}+\sqrt[{\mathrm{3}}]{\mathrm{7}}}{{sinx}+\sqrt{\mathrm{6}}}{dx}={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\sqrt[{\mathrm{3}}]{\mathrm{7}}\:\int\frac{\mathrm{1}}{{sinx}+\sqrt{\mathrm{6}}}{dx} \\ $$$$={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{7}}\:\int\frac{\mathrm{1}}{\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }+\sqrt{\mathrm{6}}}.\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:{tan}\frac{{x}}{\mathrm{2}}={t}…