Example 1. Find the indefinite integral of the vector function $\mathbf{a}(t)=$ $\mathbf{i} \cos t+\mathbf{j} e^{-t}+\mathbf{k}$
Solution. According to formula (1)

$$
\int \mathrm{a}(t) d t=1 \int \cos t d t+\mathrm{J} \int e^{-t} d t+\mathrm{k} \int d t=1 \sin t-\mathrm{j} e^{-t}+\mathrm{k} t+\mathrm{c}
$$

where $\mathbf{c}-$ is an arbitrary constant vector.

## Problems for Independent Solution

Find the integrals of the following vector-functions:
28. $\left.a(t)=t e^{t} \mid+\sin ^{2} t\right]-\frac{1}{1+t^{2}} \mathrm{k}$.
29. $\mathrm{a}(t)=\frac{t}{1+t^{2}} \mathbf{i}+t e^{t^{2}} \mathbf{j}+\cos t \mathbf{k}$.
30. $a(t)=\cos t e^{\sin t} \mathbf{i}-t \cos t^{2} \mathbf{j}+\mathbf{k}$.
31. $\mathrm{a}(t)=\frac{1}{2} t^{2} \mathbf{i}-t \sin t \mathbf{j}+2^{t} \mathbf{k}$.

Let the vector function $\mathbf{a}(t)$ be defined and continuous on some interval $\left[t_{0}, T\right]$ of the argument $t$.

Definition 3. The definite integral of the vector function $\mathbf{a}(t)$ over the interval $\left[t_{0}, T\right]$ is the limit of the vector integral sums

$$
\sigma=\sum_{k=0}^{n-1} \mathbf{a}\left(\tau_{k}\right) \Delta t_{k}, \quad \tau_{k} \in\left[t_{k}, t_{k+1}\right]
$$

as the length $\Delta t$ of the largest of the intervals $\left[t_{k}, t_{k+1}\right]$, $k=0,1, \ldots, n-1$, into which the interval $\left[t_{0}, T\right]$ is divided, tends to zero:

$$
\int_{t_{0}}^{T} \mathbf{a}(t) d t=\lim _{\Delta t \rightarrow 0} \sum_{k=0}^{n-1} \mathbf{a}\left(\tau_{k}\right) \Delta t_{k}
$$

The formula holds:

$$
\int_{t_{0}}^{T} \mathbf{a}(t) d t=\mathbf{A}(T)-\mathbf{A}\left(t_{0}\right)
$$

where $\mathbf{A}(t)$ is any antiderivative of the function $\mathbf{a}(t)$ on the interval $\left[t_{0}, T\right]$.

If

$$
\mathbf{a}(t)=a_{1}(t) \mathbf{i}+a_{2}(t) \mathbf{j}+a_{3}(t) \mathbf{k}
$$

then

$$
\int_{t_{0}}^{T} \mathbf{a}(t) d t=\mathbf{i} \int_{t_{0}}^{T} a_{1}(t) d t+\mathbf{j} \int_{t_{0}}^{T} a_{2}(t) d t+\mathbf{k} \int_{t_{0}}^{T} a_{3}(t) d t
$$