Q.$\int_{-5}^{5} \dfrac{x^7}{x^4 + 10}\, dx =$ ______.

Let $f(x) = \dfrac{x^7}{x^4 + 10}$. Since $f(-x) = -f(x)$, $f$ is odd, and $\displaystyle\int_{-a}^{a} f(x)\, dx = 0$ for an odd function. Hence the value is $0$.

Write $f(x) = \dfrac{x^7}{x^4 + 10}$ and test its symmetry:

$f(-x) = \frac{(-x)^7}{(-x)^4 + 10} = \frac{-x^7}{x^4 + 10} = -f(x).$

So $f$ is an odd function. By the property of definite integrals, …