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cos-x-7-1-3-sin-x-6-dx-




Question Number 139259 by bobhans last updated on 25/Apr/21
 ∫ ((cos x+(7)^(1/3) )/(sin x+(√6))) dx =?
$$\:\int\:\frac{\mathrm{cos}\:\mathrm{x}+\sqrt[{\mathrm{3}}]{\mathrm{7}}}{\mathrm{sin}\:\mathrm{x}+\sqrt{\mathrm{6}}}\:\mathrm{dx}\:=? \\ $$
Answered by Dwaipayan Shikari last updated on 25/Apr/21
∫((cosx+(7)^(1/3) )/(sinx+(√6)))dx=log(sinx+(√6))+(7)^(1/3)  ∫(1/(sinx+(√6)))dx  =log(sinx+(√6))+2(7)^(1/3)  ∫(1/(((2t)/(1+t^2 ))+(√6))).(dt/(1+t^2 ))           tan(x/2)=t  =log(sinx+(√6))+((2(7)^(1/3) )/( (√6)))∫(1/(t^2 +(√(2/3)) t+1))dt  =log(sinx+(√6))+((2(7)^(1/3) )/( (√6)))∫(1/((t+(1/( (√6))))^2 +(5/6)))dt  =log(sinx+(√6))+((2(7)^(1/3) )/( (√6))).((√6)/( (√5)))tan^(−1) (((√6)t+1)/( (√5)))+C  =log(sinx+(√6))+((2(7)^(1/3) )/( (√5)))tan^(−1) (((√6)tan(x/2)+1)/( (√5)))+C
$$\int\frac{{cosx}+\sqrt[{\mathrm{3}}]{\mathrm{7}}}{{sinx}+\sqrt{\mathrm{6}}}{dx}={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\sqrt[{\mathrm{3}}]{\mathrm{7}}\:\int\frac{\mathrm{1}}{{sinx}+\sqrt{\mathrm{6}}}{dx} \\ $$$$={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{7}}\:\int\frac{\mathrm{1}}{\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }+\sqrt{\mathrm{6}}}.\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:{tan}\frac{{x}}{\mathrm{2}}={t} \\ $$$$={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\frac{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{7}}}{\:\sqrt{\mathrm{6}}}\int\frac{\mathrm{1}}{{t}^{\mathrm{2}} +\sqrt{\frac{\mathrm{2}}{\mathrm{3}}}\:{t}+\mathrm{1}}{dt} \\ $$$$={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\frac{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{7}}}{\:\sqrt{\mathrm{6}}}\int\frac{\mathrm{1}}{\left({t}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\right)^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{6}}}{dt} \\ $$$$={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\frac{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{7}}}{\:\sqrt{\mathrm{6}}}.\frac{\sqrt{\mathrm{6}}}{\:\sqrt{\mathrm{5}}}{tan}^{−\mathrm{1}} \frac{\sqrt{\mathrm{6}}{t}+\mathrm{1}}{\:\sqrt{\mathrm{5}}}+{C} \\ $$$$={log}\left({sinx}+\sqrt{\mathrm{6}}\right)+\frac{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{7}}}{\:\sqrt{\mathrm{5}}}{tan}^{−\mathrm{1}} \frac{\sqrt{\mathrm{6}}{tan}\frac{{x}}{\mathrm{2}}+\mathrm{1}}{\:\sqrt{\mathrm{5}}}+{C} \\ $$

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