Answer
$$2+2 \ln (2)$$
Work Step by Step
We must integrate the integral as follows:
$$ \iint_{R} f(x,y) dA = \int_{1}^{2} \int_{-1/x}^{1/x} (x+1) \space dy \space dx \\= \int_{1}^{2}[\dfrac{x^2}{2}+x]_{-(1/x)}^{(1/x)} dx \\=\int_{1}^{2} 2+\dfrac{2}{x} dx \\=[2x+2 \ln x]_1^2 \\ 2(2-1)+2(\ln 2-\ln 1) \\=2+2 \ln (2)$$
