To evaluate the integral

$\int_{-9}^{9} x (2x^2 + 3)^3 \, dx$

using the properties of even and odd functions, let's analyze the integrand.

1. The term $x$ is an odd function, as $x(-t) = -x(t)$.
2. The expression $(2x^2 + 3)^3$ is even because $2x^2 + 3$ is an even function (as it contains only even powers of $x$), and raising an even function to an odd power still results in an even function.

Therefore, the product of an odd function, $x$, and an even function, $(2x^2 + 3)^3$, is an odd function. The integrand $x (2x^2 + 3)^3$ is thus an odd function.

Integral of an Odd Function Over Symmetric Limits

The integral of an odd function over a symmetric interval (from $-a$ to $a$) is zero:

$\int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd}.$

Therefore,

$\int_{-9}^{9} x (2x^2 + 3)^3 \, dx = 0.$

Final Answer

$\int_{-9}^{9} x (2x^2 + 3)^3 \, dx = 0.$

Would you like more details, or do you have any questions?

Here are five related questions to explore further:

1. How can you identify whether a function is even, odd, or neither?
2. What are the properties of integrals involving even and odd functions?
3. How would this integral change if the limits were not symmetric, say from $0$ to $9$?
4. What are other common applications of even and odd function properties in calculus?
5. Can you compute similar integrals with non-polynomial functions, such as trigonometric functions?

Tip: When evaluating definite integrals over symmetric limits, always check if the function is odd, as this can simplify calculations significantly.