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Question Number 194649 by SANOGO last updated on 12/Jul/23

$$calcul$$$${\int }_0^{\frac{\mathrm{\Pi }}{2}}\sqrt{4{sin}^{2}t+{cos}^{2}t}dt$$

Commented by Frix last updated on 12/Jul/23

$$\mathrm{You}\mathrm{changed}\mathrm{the}\mathrm{question}...\mathrm{but}$$$${4\sin }^{2}t+{\cos }^{2}t=1+{3\sin }^{2}t$$$$\mathrm{so}\mathrm{the}\mathrm{answer}\mathrm{is}\mathrm{the}\mathrm{same}.$$

Commented by SANOGO last updated on 12/Jul/23

$$okthankyou$$

Commented by Frix last updated on 12/Jul/23

$$\int \sqrt{1+{3\sin }^{2}t}dt=\mathrm{E}(t\mid -3)+C$$$$\left[\mathrm{complete}\mathrm{elliptic}\mathrm{integral}\mathrm{of}\mathrm{the}{2}^{\mathrm{nd}}\mathrm{kind}\right]$$$$\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}\sqrt{1+{3\sin }^{2}t}dt\approx 2.42211206$$

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