First, I would use polynomial division to simplify this fraction:

4x^2 - 4 | 3x^3 - 0x^2 - 3x + 4

= 3/4*x + 4/(4x^2 - 4)

= 3/4*x + 1/(x^2 - 1)

So now we have:

INT (3/4*x + 1/(x^2 - 1)) dx

= INT 3/4*x dx + INT 1/(x^2 - 1) dx

= 3/8*x^2 + INT 1/(x^2 - 1) dx

Use Partial fractions on the remaining integration:

INT 1/(x^2 - 1) dx

Factor the denominator:

x^2 - 1 = (x - 1)(x + 1)

Now set up your seperate fractions:

1/(x^2 - 1) = A/(x - 1) + B/(x + 1)

Multiply out the denominator (x^2 - 1) to both sides:

1 = A(x + 1) + B(x - 1) = Ax + A + Bx - B

Set terms equal on both sides (x's with x's and constants with constants)

(A + B)x = 0x --> A + B = 0 --> A = -B

(A - B) = 1 --> -B - B = 1 --> B = -1/2, A = 1/2

So the integration becomes:

INT 1/2*1/(x - 1) - 1/2*1/(x + 1) dx = 1/2ln|x - 1| - 1/2ln|x + 1|

The final answer is:

3/8*x^2 + 1/2ln|x - 1| - 1/2ln|x + 1|

Edited: My original solution had 2 as the coefficients of the ln functions when they should be 1/2.