Question
If $$f\left( { - x} \right) + f\left( x \right) = 0$$ then $$\int_a^x {f\left( t \right)dt} $$ is :
A.
an odd function
B.
an even function
C.
a periodic function
D.
none of these
Answer :
an even function
Solution :
$$\eqalign{
& {\text{Let }}\phi \left( x \right) = \int_a^x {f\left( t \right)dt.\,{\text{Then }}\phi } \left( { - x} \right) = \int_a^{ - x} {f\left( t \right)dt} \cr
& \therefore \phi \left( { - x} \right) = \int_a^x {f\left( t \right)dt} + \int_x^{ - x} {f\left( t \right)dt} \cr
& = \phi \left( x \right) + \int_x^0 {f\left( t \right)dt} + \int_0^{ - x} {f\left( t \right)dt} \cr
& = \phi \left( x \right) - \int_0^x {f\left( t \right)dt} + \int_0^x { - f\left( { - z} \right)dz,{\text{ using }}t} = - z \cr
& = \phi \left( x \right) - \int_0^x {\left\{ {f\left( t \right) + f\left( { - t} \right)} \right\}dt} \cr
& = \phi \left( x \right)\,\,\,\,\,\,\,\,\,\,\,\,\left( {\because f\left( t \right) + f\left( { - t} \right) = 0,{\text{ from the question}}} \right) \cr
& \therefore \,\,\,\phi \left( x \right){\text{ is even}}{\text{.}} \cr} $$