Definite Integration of A Non-even Non-odd Function Over A Symmetric Interval

Question

We know that when a function $f(x)$ is even then: $\int^a_{-a} f(x) \, dx =2\int^a_0 f(x) \, dx$

and when $f(x)$ is odd , then: $\int^a_{-a} f(x) \, dx=0$

My question is:

What can we say about $\int^a_{-a} f(x) \, dx$ if $f(x)$ is not even nor odd?

Answer 1

Every function can be written as the sum of an odd and even part:

$$f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2},$$

so the integral of any function over symmetric bounds is equal to the integral of it's even part; i.e.

$$\int_{-a}^a f(x)\, dx=\int_{-a}^ a\frac{f(x)+f(-x)}{2}\, dx=\int_{0}^a f(x)+f(-x)\, dx.$$

Not sure if this is what you're looking for.