Question Number 194649 by SANOGO last updated on 12/Jul/23
$${calcul}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}+{cos}^{\mathrm{2}} {t}\:}{dt} \\ $$
Commented by Frix last updated on 12/Jul/23
$$\mathrm{You}\:\mathrm{changed}\:\mathrm{the}\:\mathrm{question}...\:\mathrm{but} \\ $$$$\mathrm{4sin}^{\mathrm{2}} \:{t}\:+\mathrm{cos}^{\mathrm{2}} \:{t}\:=\mathrm{1}+\mathrm{3sin}^{\mathrm{2}} \:{t} \\ $$$$\mathrm{so}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{the}\:\mathrm{same}. \\ $$
Commented by SANOGO last updated on 12/Jul/23
$${ok}\:{thank}\:{you} \\ $$
Commented by Frix last updated on 12/Jul/23
![∫(√(1+3sin^2 t)) dt=E (t∣−3) +C [complete elliptic integral of the 2^(nd) kind] ∫_0 ^(π/2) (√(1+3sin^2 t)) dt≈2.42211206](Q194657.png)
$$\int\sqrt{\mathrm{1}+\mathrm{3sin}^{\mathrm{2}} \:{t}}\:{dt}=\mathrm{E}\:\left({t}\mid−\mathrm{3}\right)\:+{C} \\ $$$$\left[\mathrm{complete}\:\mathrm{elliptic}\:\mathrm{integral}\:\mathrm{of}\:\mathrm{the}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{kind}\right] \\ $$$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\sqrt{\mathrm{1}+\mathrm{3sin}^{\mathrm{2}} \:{t}}\:{dt}\approx\mathrm{2}.\mathrm{42211206} \\ $$