Antiderivative of Vector Function

Definition

Suppose a vector function $\mathbf{r}(t):I\to {\mathbb{R}}^n$ $\int \mathbf{r}(t)dt=\mathbf{R}(t)+\mathbf{C}$ where $\mathbf{C}$ is any constant vector.

Suppose a vector function $\mathbf{r}(t)=⟨f(t),g(t),h(t)⟩:[a,b]\to {\mathbb{R}}^3$ ${\int }_a^b\mathbf{r}(t)dt=⟨{\int }_a^{b}f(t)dt,{\int }_a^{b}g(t)dt,{\int }_a^{b}h(t)dt⟩=\left({\int }_a^{b}f(t)dt\right)\mathbf{i}+\left({\int }_a^{b}g(t)dt\right)\mathbf{j}+\left({\int }_a^{b}h(t)dt\right)\mathbf{k}$