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Find-0-2-sin-2-x-2-cosx-3-sinx-dx-




Question Number 205645 by hardmath last updated on 26/Mar/24
Find:  Ω = ∫_0 ^( (𝛑/2))  ((sin^2 x)/(2 cosx + 3 sinx)) dx = ?
$$\mathrm{Find}:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\frac{\mathrm{sin}^{\mathrm{2}} \mathrm{x}}{\mathrm{2}\:\mathrm{cosx}\:+\:\mathrm{3}\:\mathrm{sinx}}\:\mathrm{dx}\:=\:? \\ $$
Answered by Frix last updated on 26/Mar/24
t=tan (x/2) leads to  −4∫(t^2 /((t^2 +1)^2 (t^2 −3t−1)))dt  which can be solved but...  I found this:  (s^2 /(2c+3s))=((s^2 (2c−3s))/((2c+3s)(2c−3s)))=((3s^3 )/(13s^2 −4))−((2cs^2 )/(13s^2 −4))  ⇒  Ω_1 =3∫_0 ^(π/2) ((sin^3  x)/(13sin^2  x −4))dx =^(u=cos x)  3∫_0 ^1 ((u^2 −1)/(13u^2 −9))du=  =[((3u)/(13))+((2(√(13)))/(169))ln ∣(((√(13))u+3)/( (√(13))u−3))∣]_0 ^1 =  (3/(13))+((4(√(13)))/(169))(ln 2 −ln (−3+(√(13))))  Ω_2 =−2∫_0 ^(π/2) ((cos x sin^2  x)/(13sin^2  x −4))dx =^(v=sin x)  −2∫_0 ^1 (v^2 /(13v^2 −4))dv=  =[−((2v)/(13))+((2(√(13)))/(169))ln ∣(((√(13))v+2)/( (√(13))v−2))∣]_0 ^1 =  =−(2/(13))+((4(√(13)))/(169))(ln (2+(√(13))) −ln 3))  ⇒  Ω=(1/(13))+((4(√(13)))/(169))(2ln (5+(√(13))) −ln 3 −2ln 2)  ≈.232231309
$${t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\:\mathrm{leads}\:\mathrm{to} \\ $$$$−\mathrm{4}\int\frac{{t}^{\mathrm{2}} }{\left({t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left({t}^{\mathrm{2}} −\mathrm{3}{t}−\mathrm{1}\right)}{dt} \\ $$$$\mathrm{which}\:\mathrm{can}\:\mathrm{be}\:\mathrm{solved}\:\mathrm{but}… \\ $$$$\mathrm{I}\:\mathrm{found}\:\mathrm{this}: \\ $$$$\frac{{s}^{\mathrm{2}} }{\mathrm{2}{c}+\mathrm{3}{s}}=\frac{{s}^{\mathrm{2}} \left(\mathrm{2}{c}−\mathrm{3}{s}\right)}{\left(\mathrm{2}{c}+\mathrm{3}{s}\right)\left(\mathrm{2}{c}−\mathrm{3}{s}\right)}=\frac{\mathrm{3}{s}^{\mathrm{3}} }{\mathrm{13}{s}^{\mathrm{2}} −\mathrm{4}}−\frac{\mathrm{2}{cs}^{\mathrm{2}} }{\mathrm{13}{s}^{\mathrm{2}} −\mathrm{4}} \\ $$$$\Rightarrow \\ $$$$\Omega_{\mathrm{1}} =\mathrm{3}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}^{\mathrm{3}} \:{x}}{\mathrm{13sin}^{\mathrm{2}} \:{x}\:−\mathrm{4}}{dx}\:\overset{{u}=\mathrm{cos}\:{x}} {=}\:\mathrm{3}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{u}^{\mathrm{2}} −\mathrm{1}}{\mathrm{13}{u}^{\mathrm{2}} −\mathrm{9}}{du}= \\ $$$$=\left[\frac{\mathrm{3}{u}}{\mathrm{13}}+\frac{\mathrm{2}\sqrt{\mathrm{13}}}{\mathrm{169}}\mathrm{ln}\:\mid\frac{\sqrt{\mathrm{13}}{u}+\mathrm{3}}{\:\sqrt{\mathrm{13}}{u}−\mathrm{3}}\mid\right]_{\mathrm{0}} ^{\mathrm{1}} = \\ $$$$\frac{\mathrm{3}}{\mathrm{13}}+\frac{\mathrm{4}\sqrt{\mathrm{13}}}{\mathrm{169}}\left(\mathrm{ln}\:\mathrm{2}\:−\mathrm{ln}\:\left(−\mathrm{3}+\sqrt{\mathrm{13}}\right)\right) \\ $$$$\Omega_{\mathrm{2}} =−\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{cos}\:{x}\:\mathrm{sin}^{\mathrm{2}} \:{x}}{\mathrm{13sin}^{\mathrm{2}} \:{x}\:−\mathrm{4}}{dx}\:\overset{{v}=\mathrm{sin}\:{x}} {=}\:−\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{v}^{\mathrm{2}} }{\mathrm{13}{v}^{\mathrm{2}} −\mathrm{4}}{dv}= \\ $$$$=\left[−\frac{\mathrm{2}{v}}{\mathrm{13}}+\frac{\mathrm{2}\sqrt{\mathrm{13}}}{\mathrm{169}}\mathrm{ln}\:\mid\frac{\sqrt{\mathrm{13}}{v}+\mathrm{2}}{\:\sqrt{\mathrm{13}}{v}−\mathrm{2}}\mid\right]_{\mathrm{0}} ^{\mathrm{1}} = \\ $$$$\left.=−\frac{\mathrm{2}}{\mathrm{13}}+\frac{\mathrm{4}\sqrt{\mathrm{13}}}{\mathrm{169}}\left(\mathrm{ln}\:\left(\mathrm{2}+\sqrt{\mathrm{13}}\right)\:−\mathrm{ln}\:\mathrm{3}\right)\right) \\ $$$$\Rightarrow \\ $$$$\Omega=\frac{\mathrm{1}}{\mathrm{13}}+\frac{\mathrm{4}\sqrt{\mathrm{13}}}{\mathrm{169}}\left(\mathrm{2ln}\:\left(\mathrm{5}+\sqrt{\mathrm{13}}\right)\:−\mathrm{ln}\:\mathrm{3}\:−\mathrm{2ln}\:\mathrm{2}\right) \\ $$$$\approx.\mathrm{232231309} \\ $$

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