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Chapter 10: Problem 48

Find the indefinite integral. $$ \int\left(e^{t} \mathbf{i}+\sin t \mathbf{j}+\cos t \mathbf{k}\right) d t $$

Short Answer

Expert verified
The indefinite integral of the given vector field is \( e^t \mathbf{i} - \cos(t) \mathbf{j} + \sin(t) \mathbf{k} \).

Step by step solution

01

Integration of the first term

Integrate \( e^t \) w.r.t \( t \). The integral of \( e^t \) is the familiar function itself. So, the integral of \( e^t \) is \( e^t \). Thus, the first term in the vector field, upon integration, becomes \( e^t \mathbf{i} \).
02

Integration of the second term

Next, integrate \( \sin(t) \) w.r.t \( t \). The integral of \( \sin(t) \) is \( -\cos(t) \). Hence, the second term in the vector field, upon integration, becomes \( -\cos(t) \mathbf{j} \).
03

Integration of the third term

Finally, integrate \( \cos(t) \) w.r.t \( t \). The integral of \( \cos(t) \) is \( \sin(t) \). So, the third term in the vector field, upon integration, becomes \( \sin(t) \mathbf{k} \).
04

Combine all Integrals

Combine all the integrated terms to construct the integral of the entire vector field. So, the indefinite integral of the vector becomes \( e^t \mathbf{i} - \cos(t) \mathbf{j} + \sin(t) \mathbf{k} \).

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Most popular questions from this chapter

The position vector \(r\) describes the path of an object moving in the \(x y\) -plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. $$ \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j},(\sqrt{2}, \sqrt{2}) $$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(\theta)=2 \cos ^{3} \theta \mathbf{i}+3 \sin ^{3} \theta \mathbf{j} $$

Find \((a) r^{\prime \prime}(t)\) and \((b) r^{\prime}(t) \cdot r^{\prime \prime}(t)\). $$ \mathbf{r}(t)=\langle\cos t+t \sin t, \sin t-t \cos t, t\rangle $$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(\theta)=(\theta-2 \sin \theta) \mathbf{i}+(1-2 \cos \theta) \mathbf{j} $$

Evaluate the definite integral. $$ \int_{0}^{\pi / 4}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t $$
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