What are easy ways to know if the integrand is even or odd?

Question

When doing integrals I frequently miss the fact that the integrand is either even or odd which can save me time, one thing I have difficulty with is identifying if the integrand is even or odd when its not a simple integrand like cos or sin.

Are there some easy ways to identify complex integrands (granted complexity is relative to skill level) to know if they are even or odd because I really suck at it.

As an example is $x^{2n} e^{-x^2}$ for $n=1,2,3...$ even or odd?

Is there simple rules that i can identify quickly when looking at integrands?

Answer 1

If $f(x) = f(-x)$, then it is even. If $f(x) = -f(-x)$, then it is odd.

At the given example, $f(x) = x^{2n}e^{-x^{2}}$. Then we have: \begin{align*} f(-x) = (-x)^{2n}e^{-(-x)^{2}} = (-1)^{2n}x^{2n}e^{-x^{2}} = x^{2n}e^{-x^{2}} = f(x) \end{align*}

Thus we conclude that $f$ is even.

But it can happen that $f$ is neither even nor odd.

Take $f(x) = x + x^{2}$, for example.

Then $f(-x) = -x + x^{2}$ which is different from $f(x)$ as well as $-f(x)$. Hence it is not even nor odd.

Interestingly, the product of even functions is even, the product of odd functions is even and the product of odd and even functions is odd. Can you prove why this works? Similarly, the sum of even functions is even and the sum of odd functions is odd. Can you also prove why this works?

Hopefully this helps!

Answer 2

Define even and odd functions using series instead, the Taylor series of an even function around $x = 0$ includes only even powers, and that of an odd function includes only odd powers (in the radius of convergence).

Since $x^{2n}$ has only even powers regardless of $n$, $x^{2n}$ is always even, and $e^{-x^2}$ is even as $-x^2$ is even: for $f(x) = -x^2$, $f(-x) = f(x)$, so $e^{f(-x)} = e^{f(x)}$. Thus using the series idea from earlier, since the product of two even terms, $x^{2n} \cdot x^{2m} = x^{2n + 2m} = x^{2(n + m)}$ is even ($m,n \in \mathbb Z^+$), the product of two even functions is even.

Because when multiplying powers, you add their exponents, the product of an even and odd function is an odd function as even + odd = odd. This is unlike multiplication where even $\times$ odd = even.