Mathematics

Evaluate $$\displaystyle \int \frac{\sec^{2}\:x}{\sqrt{\tan^{2}\:x+4}}dx.$$


ANSWER

$$\displaystyle=\log\left | \tan\:x+\sqrt{\tan^{2}\:x+4} \right |+C$$


SOLUTION
Since derivative of $$\displaystyle \tan \:x$$ is $$\displaystyle \sec^{2}\:x.$$

Let $$\displaystyle \tan\:x=t$$ or $$\displaystyle \sec^{2}x\:dx=dt$$

$$\displaystyle \therefore \int \frac{\sec^{2}\:x}{\sqrt{\tan^{2}\:x+4}}dx=\int \frac{dt}{\sqrt{t^{2}+2^{2}}}$$

$$\displaystyle=\log\left | t+\sqrt{t^{2}+4} \right |+C$$

$$\displaystyle=\log\left | \tan\:x+\sqrt{\tan^{2}\:x+4} \right |+C$$
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Single Correct Medium Published on 17th 09, 2020
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