Posted by alsa on Saturday, June 16, 2012 at 8:56pm.
To show the results, we assume
F(x)=∫f(x)dx
then
∫f(x)dx from 0 to a is F(a)-F(0)
and
∫f(x-a)dx from 0 to a is F(0)-F(0-a)
=F(0)-F(-a)
=-F(-a)+F(0)
For the two to be equal, we require:
F(a)-F(0) = -F(-a) + F(0)
or
F(a)-F(-a) = 2F(0)
Which is not generally true. So the answer is no. A counter example is when f(x)=sin(x).
However, equality can be satisfied if F(x) is an odd function where F(0)=0 (such as sin(x)). This means that equality will hold if f(x)=±k*cos(x).
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