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Question Number 27611 by NECx last updated on 10/Jan/18

$$\int \frac{1}{2\sin {}^{2}x+4\cos {}^{2}x}dx$$

Answered by ajfour last updated on 11/Jan/18

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

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