Answer
$$\sqrt 2 $$
Work Step by Step
$$\eqalign{
& \int_{ - \pi /4}^{\pi /4} {\cos x} dx \cr
& {\text{find antiderivative use integration formulas from table 4}}{\text{.2}}{\text{.1}} \cr
& \int_{ - \pi /4}^{\pi /4} {\cos x} dx = \left( {\sin x} \right)_{ - \pi /4}^{\pi /4} \cr
& {\text{part 1 of fundamental theorem of calculus}} \cr
& = \sin \left( {\frac{\pi }{4}} \right) - \sin \left( { - \frac{\pi }{4}} \right) \cr
& {\text{recall sin}}\left( { - \theta } \right) = - \sin \theta \cr
& = \sin \left( {\frac{\pi }{4}} \right) + \sin \left( {\frac{\pi }{4}} \right) \cr
& = 2\sin \left( {\frac{\pi }{4}} \right) \cr
& {\text{simplify}} \cr
& = 2\left( {\frac{{\sqrt 2 }}{2}} \right) \cr
& = \sqrt 2 \cr} $$