Question

Solve ${\int }^{}\frac{\left(3{\tan }^{2}x+2\right){\sec }^{2}x}{{\left({\tan }^{3}x+2\tan x+5\right)}^2}dx$

Answer 1

Consider the given integral.

$I=\int \frac{(3{\tan }^{2}x+2){\sec }^{2}x}{{({\tan }^{3}x+2\tan x+5)}^2}dx$

Let $t={\tan }^{3}x+2\tan x+5$

$t={\tan }^{3}x+2\tan x+5$

$\frac{dt}{dx}=3{\tan }^{2}x{\sec }^{2}x+2{\sec }^{2}x+0$

$dt=(3{\tan }^{2}x+2){\sec }^{2}xdx$

Therefore,

$I=\int \frac{dt}{t^2}$

$I=-\frac{1}{t}+C$

On putting the value of $t$, we get

$I=-\frac{1}{({\tan }^{3}x+2\tan x+5)}+C$

Hence, this is the answer.