find the following indefinite integral. 1. ∫√(x - 2) dx

Explanation:

Step1: Use substitution

Let $u = x - 2$, then $du=dx$ and $x=u + 2$. The integral becomes $\int\sqrt{u}du=\int u^{\frac{1}{2}}du$.

Step2: Apply power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $n=\frac{1}{2}$, we have $\int u^{\frac{1}{2}}du=\frac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C=\frac{u^{\frac{3}{2}}}{\frac{3}{2}}+C=\frac{2}{3}u^{\frac{3}{2}}+C$.

Step3: Substitute back $u = x - 2$

We get $\frac{2}{3}(x - 2)^{\frac{3}{2}}+C$.

Answer:

$\frac{2}{3}(x - 2)^{\frac{3}{2}}+C$