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Chapter 10: Problem 45

Find the indefinite integral. $$ \int\left(\frac{1}{t} \mathbf{i}+\mathbf{j}-t^{3 / 2} \mathbf{k}\right) d t $$

Short Answer

Expert verified
The integral is: \( ln|t|\mathbf{i} + t\mathbf{j} - \frac{2}{5}t^{5/2}\mathbf{k} + \mathbf{C} \)

Step by step solution

01

Separate the Vector Components

Split the integral into three separate integrals, each for the i, j and k components: \[\int \frac{1}{t} dt\mathbf{i} + \int dt\mathbf{j} - \int t^{3 / 2} dt\mathbf{k}\]
02

Integrate the i Component

For the i component, we integrate \( \frac{1}{t} dt \). The integral of \( \frac{1}{t} dt \) is \( ln|t| \). So for the i component, we have:\[ln|t|\mathbf{i}\]
03

Integrate the j Component

The j component, \( \int dt \), simply integrates to \( t \). So for the j component, we have:\[t\mathbf{j}\]
04

Integrate the k Component

For the k component, we integrate \( t^{3/2} dt \). The integral of \( t^{3/2} dt \) is \( \frac{2}{5}t^{5/2} \). So for the k component, we have:\[\frac{2}{5}t^{5/2}\mathbf{k}\]
05

Combine the Components

Finally combine all the components to get the final result. Don't forget to include the integration constant \( C \) in the final result:\[ln|t|\mathbf{i} + t\mathbf{j} - \frac{2}{5}t^{5/2}\mathbf{k} + \mathbf{C}\]

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Most popular questions from this chapter

In Exercises 35 and \(36,\) use the properties of the derivative to find the following. (a) \(\mathbf{r}^{\prime}(t)\) (b) \(\mathbf{r}^{\prime \prime}(t)\) (c) \(D_{t}[\mathbf{r}(t) \cdot \mathbf{u}(t)]\) (d) \(D_{t}[3 \mathbf{r}(t)-\mathbf{u}(t)]\) (e) \(D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]\) (f) \(D_{t}[\|\mathbf{r}(t)\|], \quad t>0\) $$ \mathbf{r}(t)=t \mathbf{i}+3 t \mathbf{j}+t^{2} \mathbf{k}, \quad \mathbf{u}(t)=4 t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} $$

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In Exercises 59-66, prove the property. In each case, assume that \(\mathbf{r}, \mathbf{u},\) and \(\mathbf{v}\) are differentiable vector-valued functions of \(t,\) \(f\) is a differentiable real-valued function of \(t,\) and \(c\) is a scalar. $$ D_{t}[c \mathbf{r}(t)]=c \mathbf{r}^{\prime}(t) $$

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Find \((a) r^{\prime \prime}(t)\) and \((b) r^{\prime}(t) \cdot r^{\prime \prime}(t)\). $$ \mathbf{r}(t)=8 \cos t \mathbf{i}+3 \sin t \mathbf{j} $$
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