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Question Number 44422 by peter frank last updated on 28/Sep/18

$$usesubstitutionx=\cos {}^2\theta +3{sin}^2\theta$$$$showthat{\int }_1^3\frac{dx}{\sqrt{(x-1)(3-x)}}=\pi$$

Commented by maxmathsup by imad last updated on 28/Sep/18

$$I={\int }_0^{\frac{\pi }{2}}\frac{-2cos\theta sin\theta +6sin\theta cos\theta }{\sqrt{({cos}^2\theta -1+3{sin}^2\theta )(3-3{sin}^2\theta -{cos}^2\theta )}}d\theta$$$$={\int }_0^{\frac{\pi }{2}}\frac{4sin\theta .cos\theta }{\sqrt{2{sin}^2\theta .2{cos}^2\theta }}d\theta =2{\int }_0^{\frac{\pi }{2}}d\theta =\pi .$$

Commented by peter frank last updated on 29/Sep/18

$$thankyou$$

Commented by maxmathsup by imad last updated on 29/Sep/18

$$youarewelcomesir.$$

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