[tex]\max\{1,2^x\} = \begin{cases}1,\quad -1<x<0 \\ 2^x,\quad 0<x<1 \end{cases} \\ \\ \\\displaystyle \int_{-1}^1 \max\{1,2^x\}\, dx= \int_{-1}^0 \max\{1,2^x\}\, dx+\int_{0}^1 \max\{1,2^x\}\, dx = \\ \\ = \int_{-1}^0 1\, dx+\int_{0}^1 2^x\, dx = x\Big|_{-1}^0+ \dfrac{2^x}{\ln 2}\Bigg|_{0}^1 = 0-(-1)+\dfrac{2}{\ln 2} - \dfrac{1}{\ln 2} = \\ \\ \\=\dfrac{1+\ln 2}{\ln 2}[/tex]
