For fixed $x>0,\ x\neq 1$ the function $$ w_x(r)=\int_1^x t^{r-1} dt= \begin{cases} \frac{x^r-1}{r}\quad& r\neq 0\\ \log x\quad& r=0 \end{cases} $$ is strictly increasing.
Bernoulli inequality is equivalent to $$ w_{1+x}(r) > (<)\ w_{1+x}(1)$$