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Question Number 164672 by mathlove last updated on 20/Jan/22

Answered by Ar Brandon last updated on 20/Jan/22

$$\mathrm{\Omega }={\int }_0^{\pi }\frac{\cos x+1}{\sqrt{3+4\cos x+{\cos }^{2}x}}dx={\int }_0^{\pi }\frac{\cos x+1}{\sqrt{(\cos x+1)(\cos x+3)}}dx$$$$={\int }_0^{\pi }\sqrt{\frac{\cos x+1}{\cos x+3}}dx=\sqrt{2}{\int }_0^{\pi }\frac{\cos \left(\frac{x}{2}\right)}{\sqrt{4-{2\sin }^2\left(\frac{x}{2}\right)}}dx$$$$=2{\int }_0^{\frac{\pi }{2}}\frac{\cos x}{\sqrt{2-{\sin }^{2}x}}dx=2{\left[\arcsin \left(\frac{\sin x}{\sqrt{2}}\right)\right]}_0^{\frac{\pi }{2}}=2\left(\frac{\pi }{4}\right)=\frac{\pi }{2}$$

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