Question
$$\int_0^a {\left\{ {f\left( x \right) + f\left( { - x} \right)} \right\}dx} $$ is equal to :
A.
$$2\int_0^a {f\left( x \right)dx} $$
B.
$$\int_{ - a}^a {f\left( x \right)dx} $$
C.
0
D.
$$ - \int_{ - a}^a {f\left( { - x} \right)dx} $$
Answer :
$$\int_{ - a}^a {f\left( x \right)dx} $$
Solution :
$$\eqalign{
& I = \int_0^a {f\left( x \right)dx} + \int_0^a {f\left( { - x} \right)dx} \cr
& \,\,\,\,\, = \int_0^a {f\left( x \right)dx + } \int_0^{ - a} {f\left( z \right)d\left( { - z} \right)} \cr
& \,\,\,\,\, = \int_0^a {f\left( x \right)dx + } \int_{ - a}^0 {f\left( z \right)dz} \cr
& \,\,\,\,\, = \int_0^a {f\left( x \right)dx} + \int_{ - a}^0 {f\left( x \right)dx} \cr
& \,\,\,\,\, = \int_{ - a}^a {f\left( x \right)dx} \cr} $$