I have always found this to be very tricky
$ \begin{align*}\int_{ - 2}^0 {f(x)dx} &= - \int_{ 2}^0 {f( - u)du}\text{ What was done here?}\\ &= \int_{ 0}^2 {f( - u)du}\text{ What was done here and why?}\\&=\int_{ 0}^2 {f( u)du}\text{ Where is evenness used?}\\&=\int_{ 0}^2 {f( x)dx}\text{ Be careful in explaining this step!} \text{ ***} \end{align*}$
$ \begin{align*} \int_{ -2}^2 {f( x)dx}&=\int_{ -2}^0 {f( x)dx}+\int_{ 0}^2 {f( x)dx} \\&=\int_{ 0}^2 {f( x)dx}+\int_{ 0}^2 {f( x)dx}\\&=2\int_{ 0}^2 {f( x)dx} \end{align*}$.