Indefinite integral. Help $\int\frac{2x+3}{x^2-4}dx=\int\frac{2x+3}{(x+2)(x-2)}dx$
So...
$A=\frac{1}{4}$
$B=\frac{7}{4}$
$\frac{1}{4}\log_{10}|x+2|+\frac{7}{4}\log_{10}|x-2|+c$
The result should be: $\log_{10}|x^2-4|+\frac{3}{4}\log_{10}|\frac{x-2}{x+2}|+c$
What did I do wrong?

Your computation is correct (apart from using $\log_{10}$ instead of $\log$, with implied base $e$, or $\ln$).
Probably the book chose a different approach:
$$
\frac{2x+3}{x^2-4}=\frac{2x}{x^2-4}+\frac{3}{x^2-4}=
\frac{2x}{x^2-4}+\frac{3}{4}\frac{1}{x-2}-\frac{3}{4}\frac{1}{x+2}
$$
so integration gives
$$
\log\lvert x^2-4\rvert+\frac{3}{4}\log\left|\frac{x-2}{x+2}\right|+c
$$
However this is in no way “better” than your form for the antiderivative.
