To evaluate the integral
[tex]\[ I = \int_{-5}^{5} \frac{t^7}{t^{14} + 8} \, dt, \][/tex]
we need to notice an important property of the integrand and the limits of integration. Consider the function
[tex]\[ f(t) = \frac{t^7}{t^{14} + 8}.\][/tex]
We need to check if this function is odd or even. A function [tex]\( f(t) \)[/tex] is termed an odd function if [tex]\( f(-t) = -f(t) \)[/tex] and an even function if [tex]\( f(-t) = f(t) \)[/tex].
Let's check:
[tex]\[ f(-t) = \frac{(-t)^7}{(-t)^{14} + 8} = \frac{-t^7}{t^{14} + 8} = -\frac{t^7}{t^{14} + 8} = -f(t). \][/tex]
Since [tex]\( f(-t) = -f(t) \)[/tex], it means that [tex]\( f(t) \)[/tex] is an odd function.
Now, when we integrate an odd function over a symmetric interval [tex]\([-a, a]\)[/tex], the result is always [tex]\(0\)[/tex]. This property holds because the negative and positive parts of the function cancel each other out.
Given the integral:
[tex]\[ I = \int_{-5}^{5} \frac{t^7}{t^{14} + 8} \, dt, \][/tex]
since the integrand is [tex]\( \frac{t^7}{t^{14} + 8} \)[/tex] which is odd, the integral over the symmetric interval [tex]\([-5, 5]\)[/tex] is:
[tex]\[ I = 0. \][/tex]
Hence, the value of the integral is
[tex]\[ \boxed{0}. \][/tex]
