When integral is diverging? $$ \int f'(x)g(x)dx= f(x)g(x) - \int f(x)g'(x)dx$$
Can I conclude the folowing?
$$ if: f(x)g(x)= \pm \infty , \implies \int f'(x)g(x)dx = \pm \infty $$

I don't think you can claim that. 
Try $f(x) = \ln(x)$ and $g(x) = x$ 
Then $\int_0^1 f^{\prime}(x) g(x) dx = 1 $ but $x\ln(x)|_0^1 = - \infty$
