8. $int x^{2}(x^{3}+5)^{9}dx$

Explanation:

Step1: Use substitution

Let $u = x^{3}+5$, then $du=3x^{2}dx$, and $x^{2}dx=\frac{1}{3}du$.

Step2: Rewrite the integral

The integral $\int x^{2}(x^{3}+5)^{9}dx$ becomes $\frac{1}{3}\int u^{9}du$.

Step3: Integrate $u^{9}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\frac{1}{3}\times\frac{u^{10}}{10}+C$.

Step4: Substitute back $u$

Substitute $u = x^{3}+5$ back into the result, getting $\frac{(x^{3}+5)^{10}}{30}+C$.

Answer:

$\frac{(x^{3}+5)^{10}}{30}+C$