Question

The value of $\int \frac{dx}{\sin x\cos x+3{\cos }^{2}x}$ is

• A. $\log |\cos x+3|+c$
• B. $\log |\sin x+3|+c$
• C. $\log |\tan x+3|+c$
  
• D. $\log |\tan x-3|+c$

Answer 1

The correct option is C $\log |\tan x+3|+c$

$I=\int \frac{dx}{\sin x\cos x+3{\cos }^{2}x}$

Dividing the numerator and denominator by ${\cos }^{2}x$, we get

$I=\int \frac{{\sec }^{2}xdx}{\tan x+3}$

Let

$t=\tan x$

$dt={\sec }^{2}xdx$

$I=\int \frac{dt}{t+3}$

$I=\ln (t+3)+c$

$I=\ln (\tan x+3)+c$