10. evaluate $int_{-2}^{3}(4 - 2x^{3})dx$\n11. evaluate the integral $int\frac{xdx}{(x^{2}+4)^{4}}$

Explanation:

Step1: Use integral rules

$$\int_{-2}^{3}(4 - 2x^{3})dx=\int_{-2}^{3}4dx-\int_{-2}^{3}2x^{3}dx$$

Step2: Integrate each term

For $\int_{-2}^{3}4dx$, since $\int kdx=kx + C$ ($k$ is a constant), we have $\int_{-2}^{3}4dx=4x\big|{-2}^{3}=4\times(3 - (-2)) = 20$. For $\int{-2}^{3}2x^{3}dx$, since $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int_{-2}^{3}2x^{3}dx=2\times\frac{x^{4}}{4}\big|{-2}^{3}=\frac{1}{2}(x^{4})\big|{-2}^{3}=\frac{1}{2}(3^{4}-(-2)^{4})=\frac{1}{2}(81 - 16)=\frac{65}{2}$.

Step3: Calculate the result

$\int_{-2}^{3}(4 - 2x^{3})dx=20-\frac{65}{2}=\frac{40 - 65}{2}=-\frac{25}{2}$

Answer:

$-\frac{25}{2}$