1. **State the problem:** Evaluate the integral $$\int \frac{1}{\sqrt{4-(x+2)^2}} \, dx$$. 2. **Recall the formula:** The integral $$\int \frac{1}{\sqrt{a^2 - u^2}} \, du = \arcsin\left(\frac{u}{a}\right) + C$$ where $a$ is a constant and $u$ is a function of $x$. 3. **Identify substitution:** Let $$u = x + 2$$ so that $$du = dx$$. 4. **Rewrite the integral:** $$\int \frac{1}{\sqrt{4 - (x+2)^2}} \, dx = \int \frac{1}{\sqrt{4 - u^2}} \, du$$ 5. **Apply the formula:** Here, $a = 2$, so $$\int \frac{1}{\sqrt{4 - u^2}} \, du = \arcsin\left(\frac{u}{2}\right) + C$$ 6. **Substitute back:** Replace $u$ with $x+2$: $$\arcsin\left(\frac{x+2}{2}\right) + C$$ **Final answer:** $$\int \frac{1}{\sqrt{4-(x+2)^2}} \, dx = \arcsin\left(\frac{x+2}{2}\right) + C$$