Integration Questions

Question Number 167040 by cortano1 last updated on 05/Mar/22

$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}}\:\mathrm{dx}=? \\$$

Commented bygreogoury55 last updated on 05/Mar/22

$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sec}\:^{\mathrm{6}} {x}}{\mathrm{sec}\:^{\mathrm{6}} {x}+\mathrm{tan}\:^{\mathrm{6}} {x}}\:{dx}\: \\$$ $$\:\overset{{t}=\mathrm{tan}\:{x}} {=}\:\int_{\mathrm{0}} ^{\infty} \frac{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} +{t}^{\mathrm{6}} }\:{dt} \\$$ $$\:=\:\int_{\mathrm{0}} ^{\infty} \frac{{t}^{\mathrm{4}} +\mathrm{2}{t}^{\mathrm{2}} +\mathrm{1}}{\left(\mathrm{2}{t}^{\mathrm{2}} +\mathrm{1}\right)\left({t}^{\mathrm{4}} +{t}^{\mathrm{2}} +\mathrm{1}\right)}\:{dt} \\$$ $$=\:\frac{\mathrm{1}}{\mathrm{3}}\left[\int_{\mathrm{0}} ^{\infty} \frac{{dt}}{\mathrm{2}{t}^{\mathrm{2}} +\mathrm{1}}\:+\int_{\mathrm{0}} ^{\infty} \frac{{t}^{\mathrm{2}} +\mathrm{2}}{{t}^{\mathrm{4}} +{t}^{\mathrm{2}} +\mathrm{1}}\:{dt}\:\right] \\$$ $$=\:\frac{\pi}{\mathrm{12}}\left(\sqrt{\mathrm{2}}+\mathrm{2}\sqrt{\mathrm{3}}\:\right) \\$$