Question

What is the solution for : integrate ( 1/(9-12*x+4*x^2)) ?

Answer 1

`-1/(4x+6)+C`

Explanation:

Let's write the question formatted:

`int 1/(9-12x+4x^2) dx`

First off, we can simplify the denominator

`1/(9-12x+4x^2)=1/(2x+3)^2`

First, we are going to integrate by parts (u-sub)

`int f(u) (du)/dx dx=int f(u) du`

Let `u=2x+3, \quad (du)/dx=d/dx[2x+3]=2`
We get:

`1/2int 1/u^2 du=1/2intu^(-2)du`

Solve.

`1/2intu^(-2)du=-1/(2u)+C`

Sub `u` back.

`-1/(2u)+C=-1/(2(2x+3))+C=-1/(4x+6)+C`

`:.` `int 1/(9-12x+4x^2) dx=-1/(4x+6)+C`