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Q9  Integrate the functions \frac{\sec ^ 2 x }{\sqrt { \tan ^ 2 + 4 }}

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best_answer

The integral can be evaluated as follows

\frac{\sec ^ 2 x }{\sqrt { \tan ^ 2 + 4 }}
let \tan x =t \Rightarrow sec^2x dx =dt

\Rightarrow \int \frac{\sec^2x}{\sqrt{\tan^2x+4}}dx = \int \frac{dt}{\sqrt{t^2+2^2}}
                                               \\= \log\left | t+\sqrt{t^2+4} \right |+C\\ =\log \left | \tan x+\sqrt{ tan^2x+4} \right |+C

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manish

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