Question:
Evaluate the following integrals:
$\int \frac{\sec x \tan x}{3 \sec x+5} d x$
Solution:
Assume $3 \sec x+5=t$
$d(3 \sec x+5)=d t$
$3 \sec x \tan x=d t$
$\operatorname{Sec} x \tan x=\frac{\mathrm{dt}}{3}$
Substitute $t$ and $d t$
We get
$\Rightarrow \frac{1}{3} \int \frac{\mathrm{dt}}{\mathrm{t}}$
$\Rightarrow \frac{1}{3} \ln |\mathrm{t}|+\mathrm{c}$
But $t=3 \sec x+5$
$\therefore$ the equation becomes
$\Rightarrow \frac{1}{3} \ln |3 \sec x+5|+c$