Question

Solve $\int \frac{2\tan x{\sec }^{2}x}{{\tan }^{2}x+3\tan x+2}dx$

Answer 1

$\int \frac{2\tan x{\sec }^{2}x}{{\tan }^{2}x+3\tan x+2}dx$

Let $y=\tan x$

$\therefore dy={\sec }^{2}xdx$

$=\int \frac{2y.{\sec }^{2}x}{y^2+3y+2}\frac{dy}{{\sec }^{2}x}$

$=\int \frac{2y}{y^2+2y+y+2}dy$

$=\int \frac{2y}{(y+1)(y+2)}dy$

$=2\int \frac{y}{(y+1)(y+2)}dy$

$=2\int \frac{2(y+1)-(y+2)}{(y+1)(y+2)}dy$

$2\int \frac{2}{y+2}-\frac{1}{y+1}dy$

$=2[2\ln (y+2)-\ln (y+1)]+C$

Now, $y=\tan x$

$=2\ln \left[\frac{{(2+\tan x)}^2}{(1+\tan x)}\right]+C$