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calcul-0-2-4sin-2-t-cos-2-t-dt-




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

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