# How do you integrate f(x)=(3x^2-x+2)/(4x^2+5) using the quotient rule?

Oct 24, 2016

There is no quotient rule for integrating. To differentiate, see below.

#### Explanation:

We can differentiate using the quotient rule (for derivatives).

For $f \left(x\right) = \frac{u}{v}$ we have, $f ' \left(x\right) = \frac{u ' v - u v '}{v} ^ 2$.

In this function, $u = 3 {x}^{2} - x + 2$, so $u ' = 6 x - 1$

and $v = 4 {x}^{2} + 5$, so $v ' = 8 x$.

So, we get

$f ' \left(x\right) = \frac{\left(6 x - 1\right) \left(4 {x}^{2} + 5\right) - \left(3 {x}^{2} - x + 2\right) \left(8 x\right)}{4 {x}^{2} + 5} ^ 2$

$= \frac{4 {x}^{2} + 14 x - 5}{4 {x}^{2} + 5} ^ 2$