Question

use symmetry properties to evaluate the definite integral.
int_{-9}^{9} \frac{t^{5}}{t^{10}+3} dt

Explanation:

Step1: Define the function

Let $f(t)=\frac{t^{5}}{t^{10}+3}$.

Step2: Check for odd - function property

Find $f(-t)$:
\[

$$\begin{align*} f(-t)&=\frac{(-t)^{5}}{(-t)^{10}+3}\\ &=\frac{-t^{5}}{t^{10}+3}\\ &=-f(t) \end{align*}$$

\]
So, $y = f(t)$ is an odd - function.

Step3: Apply the definite - integral property for odd functions

If $y = f(t)$ is an odd function and the integral is of the form $\int_{-a}^{a}f(t)dt$, then $\int_{-a}^{a}f(t)dt = 0$. Here $a = 9$ and $f(t)=\frac{t^{5}}{t^{10}+3}$. So, $\int_{-9}^{9}\frac{t^{5}}{t^{10}+3}dt=0$.

Answer:

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