Question

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

• A. $=\log |\tan x+\sqrt{{\tan }^{2}x+4}|+C$
• B. $=\log |\tan x+\sqrt{4{\tan }^{2}x+4}|+C$
• C. $=\log |\tan x-\sqrt{{\tan }^{2}x+4}|+C$
• D. $=\log |\tan x+\sqrt{{\tan }^{2}x-4}|+C$

Answer 1

The correct option is A $=\log |\tan x+\sqrt{{\tan }^{2}x+4}|+C$

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

Let $\tan x=t$ or ${\sec }^{2}xdx=dt$

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

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

$=\log |\tan x+\sqrt{{\tan }^{2}x+4}|+C$