Question Number 126753 by bemath last updated on 24/Dec/20 $$\:\:\int\:\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{3sin}\:{x}+\mathrm{4cos}\:{x}+\mathrm{1}}\:{dx}\: \\ $$ Answered by Ar Brandon last updated on 24/Dec/20 $$\mathrm{sinx}+\mathrm{cosx}=\lambda\left(\mathrm{3sinx}+\mathrm{4cosx}+\mathrm{1}\right)+\mu\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{3sinx}+\mathrm{4cosx}+\mathrm{1}\right)\right\}+\gamma \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\lambda\left(\mathrm{3sinx}+\mathrm{4cosx}+\mathrm{1}\right)+\mu\left(\mathrm{3cosx}−\mathrm{4sinx}\right)+\gamma \\ $$$$\begin{cases}{\mathrm{3}\lambda−\mathrm{4}\mu=\mathrm{1}}\\{\mathrm{4}\lambda+\mathrm{3}\mu=\mathrm{1}}\\{\lambda+\gamma=\mathrm{0}}\end{cases}\Rightarrow\begin{cases}{\mathrm{25}\mu=−\mathrm{1}}\\{\mathrm{25}\lambda=\mathrm{7}}\\{\gamma=−\lambda}\end{cases}…
Question Number 126749 by bemath last updated on 24/Dec/20 Commented by MJS_new last updated on 24/Dec/20 $${t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\:\mathrm{leads}\:\mathrm{to} \\ $$$$\mathrm{2}\int\frac{\left({t}+\mathrm{1}\right)^{\mathrm{2}} }{{t}^{\mathrm{4}} +\mathrm{10}{t}^{\mathrm{2}} +\mathrm{1}}{dt} \\ $$$$\mathrm{now}\:\mathrm{find}\:\mathrm{the}\:\mathrm{2}\:\mathrm{square}\:\mathrm{factors}\:\mathrm{of}\:\mathrm{the}\:\mathrm{denominator} \\…
Question Number 126746 by bemath last updated on 24/Dec/20 $$\:\:\int\:\frac{{dx}}{\mathrm{tan}\:^{\mathrm{5}} {x}+\mathrm{1}}\:? \\ $$ Commented by MJS_new last updated on 24/Dec/20 $${t}=\mathrm{tan}\:{x}\:\mathrm{leads}\:\mathrm{to} \\ $$$$\int\frac{{dt}}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)\left({t}^{\mathrm{5}} +\mathrm{1}\right)}…
Question Number 192280 by Mingma last updated on 14/May/23 Answered by witcher3 last updated on 14/May/23 $$\Omega=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{4}} −\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$=\int_{\mathrm{0}} ^{\infty}…
Question Number 61208 by Tawa1 last updated on 30/May/19 Commented by maxmathsup by imad last updated on 30/May/19 $${its}\:{only}\:{a}\:{try}\:{let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:\:{with}\:{t}\:\geqslant\mathrm{0} \\ $$$${we}\:{have}\:{f}^{'}…
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Question Number 126710 by slahadjb last updated on 23/Dec/20 $$\int{ln}\left({x}\right){e}^{{x}^{\mathrm{2}} } {dx}\:\:\:\:\:\:??? \\ $$ Answered by Dwaipayan Shikari last updated on 23/Dec/20 $$\int{e}^{{x}^{\mathrm{2}} } {log}\left({x}\right){dx}\:\:\:\:\:{x}^{\mathrm{2}}…
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