Question-164672

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

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

Ω = \int_{0}^{π} \frac{\cos x + 1}{\sqrt{3 + 4 \cos x + \cos^{2} x}} d x = \int_{0}^{π} \frac{\cos x + 1}{\sqrt{(\cos x + 1) (\cos x + 3)}} d x

= \int_{0}^{π} \sqrt{\frac{\cos x + 1}{\cos x + 3}} d x = \sqrt{2} \int_{0}^{π} \frac{\cos (\frac{x}{2})}{\sqrt{4 - {2 \sin}^{2} (\frac{x}{2})}} d x

= 2 \int_{0}^{\frac{π}{2}} \frac{\cos x}{\sqrt{2 - \sin^{2} x}} d x = 2 {[\arcsin (\frac{\sin x}{\sqrt{2}})]}_{0}^{\frac{π}{2}} = 2 (\frac{π}{4}) = \frac{π}{2}