TITLE: Using partial fraction techniques to work out following intergrals
QUESTION [1 upvotes]: Hello I am studying for a mock test that is coming up, a question very similar to this one will be on this test, I have no idea how to complete this type of question, I have been given some vague notes about how I should turn it into a arctan function (??? I have no idea how to)
Here is the question:
$$I=\int\frac{5x^{2}-10x}{(9x^{2}-16)(x^{2}+1)}\:\mathrm{d}x$$
I've 'Simplified' down to: 
$$\frac{5x^2-10x}{(3x-4)(3x-4)(x^2+1)}$$
Here is the stage I have gotten to so far:
$$I = \int\left(\frac{A}{3x-4} + \frac{B}{3x-4} + Cx+\frac{D}{x^2+1}\right)\:\mathrm{d}x$$

REPLY [0 votes]: Hint. 
You may want to use

$$
\int\frac1{a x+b}dx=\frac1a\log \left| a x+b\right|+C \quad (a\neq0)\tag1
$$ 

and 

$$
\int\frac{x}{x^2+1}dx=\frac12\int\frac{2x}{x^2+1}dx=\frac12\int\frac{(x^2+1)'}{x^2+1}dx=\frac12\log \left( x^2+1\right)+C \tag2
$$ 

and

$$
\begin{align}
\int \frac{1}{x^2+1}dx&=\arctan x +C \tag3
\end{align}
$$
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