Can you give the example of an odd integrand and prove that it is odd?

Example of an odd integrand and it's proof.

Certainly! Let's consider the function $f(x) = x^3$. We'll integrate this function over a symmetric interval, say $[-a, a]$, and then show that the result is zero, proving that $f(x)$ is an odd function.

The integral is given by:

$\int_{-a}^{a} x^3 \,dx$

To evaluate this integral, we'll use the properties of odd functions. An odd function $f(x)$ satisfies $f(-x) = -f(x)$. Let's evaluate $f(-x)$ for $f(x) = x^3$:

$f(-x) = (-x)^3 = -x^3$

Now, compare $f(-x)$ with $-f(x)$:

$-f(x) = -x^3$

As we can see, $f(-x) = -f(x)$. This property holds for all values of $x$.

Now, let's calculate the integral:

$\int_{-a}^{a} x^3 \,dx = \frac{1}{4}x^4 \Big|_{-a}^{a}$

$= \frac{1}{4}(a^4 - (-a)^4)$

$= \frac{1}{4}(a^4 - a^4)$

$= \frac{1}{4}(0)$

$= 0$

The result is zero, which means that the integral of the odd function $f(x) = x^3$ over the symmetric interval $[-a, a]$ is zero. This demonstrates that $f(x) = x^3$ is an odd function.