Question

Integrate the following functions w.r.t. $x$ :
$\frac{1+{\tan }^{2}x}{\sqrt{{\tan }^{2}x+3}}$

Answer 1

$\int \frac{1+{\tan }^{2}x}{\sqrt{{\tan }^{2}x+3}}dx=\int \frac{{\sec }^{2}xdx}{\sqrt{(\sqrt{3}{)}^2+{\tan }^{2}x}}$
Let $\tan x=t$
$\begin{array}{cc}{\sec }^{2}xdx& =dt\\ & =\int \frac{dt}{\sqrt{(\sqrt{3}{)}^2+t^2}}\\ & =\log |t+\sqrt{t^2+3}|+C\\ & =\log |\tan x+\sqrt{{\tan }^{2}x+3}|+C\end{array}$