Write the following expression $as$ a single definite integral $of$ the form ${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx.$

${\displaystyle \underset{-5}{\overset{1}{}}}f(x)dx+{\displaystyle \underset{1}{\overset{2}{}}}f(x)dx-{\displaystyle \underset{-5}{\overset{-2}{}}}f(x)dx$

Step $1$

$We$ know that for $abc,{\displaystyle \underset{a}{\overset{b}{}}}f(x)dx+{\displaystyle \underset{b}{\overset{c}{}}}f(x)dx={\displaystyle \underset{a}{\overset{c}{}}}f(x)dx.$

Using the first and second integrals, since $we$ have $-512,$

${\displaystyle \underset{-5}{\overset{1}{}}}f(x)dx+{\displaystyle \underset{1}{\overset{2}{}}}f(x)dx={\displaystyle \underset{-5}{\overset{2}{}}}f(x)dx$

Step $2$

$We$ also know that ${\displaystyle \underset{b}{\overset{a}{}}}f(x)dx=-{\displaystyle \underset{a}{\overset{b}{}}}f(x)dx.$ Therefore, $we$ can say the following.

$-{\displaystyle \underset{-5}{\overset{-2}{}}}f(x)dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$