1. **Problem Statement:** Define the indefinite integral of a vector-valued function and provide an example. 2. **Definition:** The indefinite integral of a vector-valued function \( \mathbf{F}(t) = \langle f_1(t), f_2(t), \ldots, f_n(t) \rangle \) is a vector function \( \mathbf{G}(t) = \langle g_1(t), g_2(t), \ldots, g_n(t) \rangle \) such that $$\mathbf{G}'(t) = \mathbf{F}(t)$$ where each component \( g_i(t) \) is an antiderivative of \( f_i(t) \). 3. **Formula:** For each component, $$g_i(t) = \int f_i(t) \, dt + C_i$$ where \( C_i \) is an arbitrary constant of integration. 4. **Explanation:** The indefinite integral of a vector-valued function is found by integrating each component function separately. The result is a vector of antiderivatives plus a vector of constants. 5. **Example:** Consider \( \mathbf{F}(t) = \langle 2t, \cos t, e^t \rangle \). Integrate each component: - \( \int 2t \, dt = t^2 + C_1 \) - \( \int \cos t \, dt = \sin t + C_2 \) - \( \int e^t \, dt = e^t + C_3 \) So, $$\mathbf{G}(t) = \langle t^2 + C_1, \sin t + C_2, e^t + C_3 \rangle$$ This \( \mathbf{G}(t) \) is the indefinite integral of \( \mathbf{F}(t) \). 6. **Summary:** The indefinite integral of a vector-valued function is the vector of indefinite integrals of its components, each with its own constant of integration.