What's wrong with my answer? $\int \frac{3x^2-2}{x^2-4x-12} \mathrm dx$

Question

Here is the original problem: $\int \frac{3x^2-2}{x^2-4x-12}\ \mathrm dx$

After doing polynomial division and factoring the denominator I got this:

$$\int 3 + \frac{12x+36}{(x-6)(x+2)}\ \mathrm dx$$

Then using partial fraction decomposition I got the following:

$$\int 3\ \mathrm dx+ \frac{27}{2}\int \frac{\ \mathrm dx}{x-6} -\frac{3}{2}\int \frac{\ \mathrm dx}{x+2}$$

For the final answer I got this:

$$3x+\frac{27}{2}ln|x-6|-\frac{3}{2}ln|x+2|+C$$

But it says my answer is incorrect. Can you all spot my error or should I provide more details?

Answer 1

You did the calculation wrong. $$\frac{3x^2-2}{x^2-4x-12} = \frac{3(x^2-4x-12)-2+12x+36}{x^2-4x-12} = 3+\frac{12x+34}{x^2-4x-12}$$

Now, try to evaluate your integral