Question

1. $\int (x - 2)^8\cdot6x dx$

Explanation:

Step1: Use substitution

Let \(u = x - 2\), then \(x=u + 2\) and \(dx=du\).
The integral becomes \(\int u^{8}\cdot6(u + 2)du=6\int(u^{9}+2u^{8})du\).

Step2: Integrate term - by - term

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\)
\(6\int(u^{9}+2u^{8})du=6(\frac{u^{10}}{10}+\frac{2u^{9}}{9})+C\).

Step3: Substitute back

Substitute \(u=x - 2\) back in:
\(6(\frac{(x - 2)^{10}}{10}+\frac{2(x - 2)^{9}}{9})+C=\frac{3(x - 2)^{10}}{5}+\frac{4(x - 2)^{9}}{3}+C\).

Answer:

\(\frac{3(x - 2)^{10}}{5}+\frac{4(x - 2)^{9}}{3}+C\)