X2 T05 08 odd and even functions (2010)

Even integrand over a symmetric integral
Even integrand over a symmetric integral

Odd & Even Functions
(1) Even f x f x
a a
f x dx 2 f x dx
a 0

Odd & Even Functions
(1) Even f x f x
a a
f x dx 2 f x dx
a 0
ca ca
NOTE: horizontal shift
f x c dx 2 f x c dx
ca c

Odd & Even Functions
(1) Even f x f x
a a
f x dx 2 f x dx
a 0
ca ca
NOTE: horizontal shift
f x c dx 2 f x c dx
ca c
(2) Odd f x f x

Odd & Even Functions
(1) Even f x f x
a a
f x dx 2 f x dx
a 0
ca ca
NOTE: horizontal shift
f x c dx 2 f x c dx
ca c
(2) Odd f x f x
a
f x dx 0
a

Odd & Even Functions
(1) Even f x f x
a a
f x dx 2 f x dx
a 0
ca ca
NOTE: horizontal shift
f x c dx 2 f x c dx
ca c
(2) Odd f x f x
a
f x dx 0
a
ca
NOTE: horizontal shift
f x c dx 0
ca

Odd & Even Functions
(1) Even f x f x
a a
f x dx 2 f x dx
a 0
ca ca
NOTE: horizontal shift
f x c dx 2 f x c dx
ca c
(2) Odd f x f x
a
f x dx 0
a
ca
NOTE: horizontal shift
f x c dx 0
ca
a a
f x dx f a x dx
(3)
0 0

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