Question

Is Integral of f(x) w.r.t.x from 0 to a;equal to Integral of f(x-a) w.r.t.x from 0 to a?

Answer 1

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).