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Question Number 27612 by NECx last updated on 10/Jan/18

$$\int \frac{1}{3+{cos}^{2}x}dx$$

Answered by ajfour last updated on 11/Jan/18

$$=\int \frac{\sec {}^{2}xdx}{3\sec {}^{2}x+1}=\int \frac{d\left(\tan x\right)}{4+3\tan {}^{2}x}$$$$witht=\tan xitbecomes$$$$\int \frac{dt}{3t^2+4}=\frac{1}{3}\int \frac{dt}{t^2+{\left(2/\sqrt{3}\right)}^2}$$$$=\frac{1}{2\sqrt{3}}{\tan }^{-1}\left(\frac{t\sqrt{3}}{2}\right)$$$$=\frac{1}{2\sqrt{3}}{\tan }^{-1}\left(\frac{\sqrt{3}\tan x}{2}\right)+c.$$

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