C-0-pi-2-cos-2-x-4sin-2-x-cos-2-x-dx-

Question Number 178988 by cortano1 last updated on 23/Oct/22

C = \underset{0}{\int^{π / 2}} \frac{\cos^{2} x}{4 \sin^{2} x + \cos^{2} x} dx = ?

Answered by qaz last updated on 23/Oct/22

C = \int_{0}^{π / 2} \frac{\cos^{2} x}{4 {s i n}^{2} x + {c o s}^{2} x} d x

= \int_{0}^{π / 2} \frac{\sec^{2} x d x}{(4 \tan^{2} x + 1) (1 + \tan^{2} x)}

= \int_{0}^{\infty} \frac{d x}{(4 x^{2} + 1) (1 + x^{2})}

= \frac{1}{3} \int_{0}^{\infty} (\frac{4}{1 + 4 x^{2}} - \frac{1}{1 + x^{2}}) d x

= \frac{2}{3} \arctan 2 x - \frac{1}{3} \arctan x ∣_{0}^{\infty}

= \frac{π}{6}

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