1. **Stating the problem:** Evaluate the integral $$\int \frac{bx + c}{x^2 + 2} \, dx$$ and identify the correct form among the given options. 2. **Formula and rules:** We use the method of splitting the integral into two parts: $$\int \frac{bx}{x^2 + 2} \, dx + \int \frac{c}{x^2 + 2} \, dx$$ Recall that: - $$\int \frac{x}{x^2 + a^2} \, dx = \frac{1}{2} \ln(x^2 + a^2) + C$$ - $$\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$$ 3. **Intermediate work:** Split the integral: $$\int \frac{bx}{x^2 + 2} \, dx = b \int \frac{x}{x^2 + 2} \, dx = b \cdot \frac{1}{2} \ln(x^2 + 2) + C = \frac{b}{2} \ln(x^2 + 2) + C$$ $$\int \frac{c}{x^2 + 2} \, dx = c \int \frac{1}{x^2 + (\sqrt{2})^2} \, dx = \frac{c}{\sqrt{2}} \arctan\left(\frac{x}{\sqrt{2}}\right) + C$$ 4. **Combine results:** $$\int \frac{bx + c}{x^2 + 2} \, dx = \frac{b}{2} \ln(x^2 + 2) + \frac{c}{\sqrt{2}} \arctan\left(\frac{x}{\sqrt{2}}\right) + C$$ 5. **Answer:** The correct choice is (c).