Answer
$$\sqrt 2 $$
Work Step by Step
$$\eqalign{
& \int_{ - \pi /4}^{\pi /4} {\cos xdx} \cr
& {\text{ the function }}\cos x{\text{ is even}} \cr
& {\text{Use the theorem 5}}{\text{.4}} \cr
& {\text{If }}f\left( x \right){\text{ is even}}{\text{, }}\int_{ - a}^a {f\left( x \right)dx} = 2\int_0^a {f\left( x \right)dx} \cr
& then \cr
& = 2\int_0^{\pi /4} {\cos xdx} \cr
& {\text{integrating}} \cr
& = 2\left. {\left( {\sin x} \right)} \right|_0^{\pi /4} \cr
& {\text{using the fundamental theorem}} \cr
& = 2\left( {\sin \frac{\pi }{4} - \sin 0} \right) \cr
& = 2\left( {\frac{{\sqrt 2 }}{2} - 0} \right) \cr
& = \sqrt 2 \cr} $$
