this problem is about applying power rule: $int u(5u^{2}-9)^{14}du$

Explanation:

Step1: Use substitution

Let $t = 5u^{2}-9$, then $dt=10u\ du$, and $u\ du=\frac{1}{10}dt$.

Step2: Rewrite the integral

The integral $\int u(5u^{2}-9)^{14}du$ becomes $\frac{1}{10}\int t^{14}dt$.

Step3: Apply power - rule

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). So, $\frac{1}{10}\int t^{14}dt=\frac{1}{10}\times\frac{t^{15}}{15}+C$.

Step4: Substitute back

Substitute $t = 5u^{2}-9$ back into the result. We get $\frac{(5u^{2}-9)^{15}}{150}+C$.

Answer:

$\frac{(5u^{2}-9)^{15}}{150}+C$