Let $I$ = the value of integration and $p$ = probability distribution.
The estimator is denoted as $\left\langle I \right >$ and is $$\left\langle I \right >=\frac{1}{N}\sum_{i=1}^{N}\frac{f(x_i)}{p(x_i)}$$
The expected value of this estimator is computed as follows:
$$E[\left\langle I \right >] = E[\frac{1}{N}\sum_{i=1}^{N}\frac{f(x_i)}{p(x_i)}]$$
$$\qquad \qquad \space \space \space \space = \frac{1}{N}E[\sum_{i=1}^{N}\frac{f(x_i)}{p(x_i)}]\cdots(1)$$
$$ \qquad \qquad\qquad\quad= \frac{1}{N}N\int\frac{f(x)}{p(x)}p(x)dx\cdots(2)$$
$$ = \int f(x)dx \space \space$$ $$ = I\qquad\qquad \space$$
I wonder why the two equation, $(1)$ and $(2)$, are equal??