# power series representation theorem help

I am having trouble trying to understand a step in this theorem. Basically how do they justify $\int_{1}^{1+x} \frac{(1-t)^n}{t} dt\leq \int_{1}^{1+x} (t-1)^n dt$. I tried solving the left side integral but it is pretty ugly. I'm sure there is something obvious i'm not seeing. Clarifying this step will be great thanks.

If $f(x) \leq g(x)$ on $[a,b]$ for all $x$, then $\int_a^b f(t) dt \leq \int_a^b g(t) dt$. In this case, $\frac{(1 - t)^n}{t} \leq (t - 1)^n$ on $[1, 1 + x]$ since $t \geq 1$ on this interval.