UNKNOWN Questions

Question Number 55089 by eso last updated on 17/Feb/19

$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }\:{dx}\:= \\$$

Commented bymaxmathsup by imad last updated on 17/Feb/19

$${let}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\:\:{changement}\:{x}\:={tant}\:{give}\: \\$$ $${I}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{\mathrm{1}+{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+{tan}^{\mathrm{2}} {t}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{dt}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{dt}}{\left(\mathrm{1}+{tan}^{\mathrm{2}} {t}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} }\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\left({cos}^{\mathrm{2}} {t}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:{dt} \\$$ $$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cost}\:{dt}\:=\left[{sint}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\$$