1. **State the problem:** We are given a differential equation $\frac{dy}{dx} = f(x)$ and a slope field. We need to find which function could be $\int f(x) \, dx$ from the given options. 2. **Recall the relationship:** The slope field represents the slopes $\frac{dy}{dx} = f(x)$ at various points. The integral $\int f(x) \, dx$ gives the family of functions whose derivative is $f(x)$. 3. **Analyze the options:** The options are: - a. $\sin x + C$ - b. $\cos x + C$ - c. $-\sin x + C$ - d. $-\cos x + C$ - e. $\frac{\sin^2 x}{2} + C$ 4. **Find $f(x)$ for each option:** Since $\frac{d}{dx} \int f(x) dx = f(x)$, differentiate each option: - a. $\frac{d}{dx} (\sin x + C) = \cos x$ - b. $\frac{d}{dx} (\cos x + C) = -\sin x$ - c. $\frac{d}{dx} (-\sin x + C) = -\cos x$ - d. $\frac{d}{dx} (-\cos x + C) = \sin x$ - e. $\frac{d}{dx} \left( \frac{\sin^2 x}{2} + C \right) = \sin x \cos x$ 5. **Match with slope field:** The slope field shape is divided into 4 sections by vertical and horizontal lines crossing at the center, suggesting slopes depend on $\sin x$ or $\cos x$. 6. **Conclusion:** The slope field likely corresponds to $f(x) = \sin x$, so the integral is $-\cos x + C$ (option d). **Final answer:** $-\cos x + C$ (option d)