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Question Number 209662 by depressiveshrek last updated on 18/Jul/24

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

Answered by ARUNG_Brandon_MBU last updated on 18/Jul/24

$$I=\int \frac{dx}{{\sin }^{2}x(1+{\cos }^{2}x)}$$$$=\int \frac{{\tan }^{2}x+1}{{\tan }^{2}x({\tan }^{2}x+2)}\left({\sec }^{2}xdx\right)$$$$=\frac{1}{2}\int \frac{({\tan }^{2}x+2)+{\tan }^{2}x}{{\tan }^{2}x({\tan }^{2}x+2)}d\left(\tan x\right)$$$$=-\frac{1}{2\tan x}+\frac{1}{2\sqrt{2}}\arctan \left(\frac{\tan x}{\sqrt{2}}\right)+C$$

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