Question Number 68116 by aliesam last updated on 05/Sep/19
$$\int\frac{{dx}}{{sin}\mathrm{2}{x}−{sec}\left({x}\right)} \\ $$
Commented by MJS last updated on 06/Sep/19
$$\mathrm{look}\:\mathrm{at}\:\mathrm{question}\:\mathrm{68141} \\ $$
Answered by MJS last updated on 05/Sep/19
![∫(dx/(sin 2x −sec x))= [t=sin x → dt=(dt/(cos x))] =−(1/2)∫(dt/(t^3 −t+(1/2)))=−(1/2)∫(dt/((t−α)(t−β)(t+γ))) now decompose and integrate at the end solve t^3 −t+(1/2)=0 to get the constant factors](https://www.tinkutara.com/question/Q68118.png)
$$\int\frac{{dx}}{\mathrm{sin}\:\mathrm{2}{x}\:−\mathrm{sec}\:{x}}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{sin}\:{x}\:\rightarrow\:{dt}=\frac{{dt}}{\mathrm{cos}\:{x}}\right] \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dt}}{{t}^{\mathrm{3}} −{t}+\frac{\mathrm{1}}{\mathrm{2}}}=−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dt}}{\left({t}−\alpha\right)\left({t}−\beta\right)\left({t}+\gamma\right)} \\ $$$$\mathrm{now}\:\mathrm{decompose}\:\mathrm{and}\:\mathrm{integrate} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{end}\:\mathrm{solve}\:{t}^{\mathrm{3}} −{t}+\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0}\:\mathrm{to}\:\mathrm{get}\:\mathrm{the} \\ $$$$\mathrm{constant}\:\mathrm{factors} \\ $$
Commented by mind is power last updated on 05/Sep/19
$${cardan} \\ $$$$ \\ $$
Commented by MJS last updated on 05/Sep/19
$$\mathrm{what}\:\mathrm{does}\:“\mathrm{cardan}''\:\mathrm{mean}? \\ $$
Commented by mind is power last updated on 05/Sep/19
$${cardan}\:{is}\:{one}\:{way}\:{to}\:{solve}\:\:{algebric}\:{equation}\:{of}\:{degre}\:\mathrm{3}\:{sorry}\:{fir}\:{my}\:{english}\: \\ $$$$ \\ $$
Commented by MJS last updated on 06/Sep/19
$$\mathrm{of}\:\mathrm{course}!\:\mathrm{you}\:\mathrm{mean}\:\mathrm{Cardano}'\mathrm{s}\:\mathrm{method},\:\mathrm{you} \\ $$$$\mathrm{are}\:\mathrm{right}.\:\mathrm{I}\:\mathrm{read}\:“\mathrm{cardan}''\:\mathrm{and}\:\mathrm{the}\:\mathrm{only}\:\mathrm{thing} \\ $$$$\mathrm{that}\:\mathrm{came}\:\mathrm{to}\:\mathrm{my}\:\mathrm{mind}\:\mathrm{was}\:\mathrm{the}\:“\mathrm{cardan}\:\mathrm{shaft}\: \\ $$$$\mathrm{drive}''…\:\mathrm{haha},\:\mathrm{blame}\:\mathrm{it}\:\mathrm{on}\:\mathrm{me}… \\ $$
Commented by mind is power last updated on 05/Sep/19
$${yeah}\:{sorry}\:{im}\:{french}\:{my}\:{english}\:{is}\:{not}\:{verry}\:{good} \\ $$
Commented by MJS last updated on 06/Sep/19
no problem, I'm from Austria... I had 4 years of French in school but that had been 30 years ago... Je peux encore comprendre partiellement le français écrit, mais je ne comprends plus la langue parlée ...