Integral calculus
posted by alsa on .
Is Integral of f(x) w.r.t.x from 0 to a;equal to Integral of f(xa) w.r.t.x from 0 to a?

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(xa)dx from 0 to a is F(0)F(0a)
=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).