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Chapter 12: Problem 51

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=e^{3 t} \mathbf{i}+\frac{1}{1+t^{2}} \mathbf{j}-\frac{1}{\sqrt{2 t}} \mathbf{k}$$

Short Answer

Expert verified
Answer: The indefinite integral of the given vector function is \(\int \mathbf{r}(t) \, dt = \left(\frac{1}{3}e^{3t} + C_1\right)\mathbf{i} + \left(\arctan(t) + C_2\right)\mathbf{j} + \left(-\sqrt{2t} + C_3\right)\mathbf{k}\).

Step by step solution

01

Identify the Components

First, identify the components of the vector function: $$\mathbf{r}(t) = e^{3t}\mathbf{i} + \frac{1}{1+t^2}\mathbf{j} - \frac{1}{\sqrt{2t}}\mathbf{k}$$ The components are: - \(f(t) = e^{3t}\) - \(g(t) = \frac{1}{1+t^2}\) - \(h(t) = -\frac{1}{\sqrt{2t}}\)
02

Integrate the Components

Integrate each component with respect to \(t\): 1. \(\int f(t) \, dt= \int e^{3t} \, dt\) 2. \(\int g(t) \, dt= \int \frac{1}{1+t^2} \,dt\) 3. \(\int h(t) \, dt= \int -\frac{1}{\sqrt{2t}} \, dt\)
03

Find the Indefinite Integrals of the Components

Find the indefinite integral for each component: 1. \(\int e^{3t} \, dt = \frac{1}{3}e^{3t} + C_1\) 2. \(\int \frac{1}{1+t^2} \, dt = \arctan(t) + C_2\) 3. \(\int -\frac{1}{\sqrt{2t}} \, dt = -\sqrt{2t} + C_3\)
04

Write the Vector Function

Combine the integrals of the components to create the integral vector function: $$\int \mathbf{r}(t) \, dt = \left(\frac{1}{3}e^{3t} + C_1\right)\mathbf{i} + \left(\arctan(t) + C_2\right)\mathbf{j} + \left(-\sqrt{2t} + C_3\right)\mathbf{k}$$ The indefinite integral of the given vector function is: $$\int \mathbf{r}(t) \, dt = \left(\frac{1}{3}e^{3t} + C_1\right)\mathbf{i} + \left(\arctan(t) + C_2\right)\mathbf{j} + \left(-\sqrt{2t} + C_3\right)\mathbf{k}$$

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Most popular questions from this chapter

\(\mathbb{R}^{3}\) Consider the vectors \(\mathbf{I}=\langle 1 / 2,1 / 2,1 / \sqrt{2}), \mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}, 0\rangle,\) and \(\mathbf{K}=\langle 1 / 2,1 / 2,-1 / \sqrt{2}\rangle\) a. Sketch I, J, and K and show that they are unit vectors. b. Show that \(\mathbf{I}, \mathbf{J},\) and \(\mathbf{K}\) are pairwise orthogonal. c. Express the vector \langle 1,0,0\rangle in terms of \(\mathbf{I}, \mathbf{J},\) and \(\mathbf{K}\).

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and fare real numbers. It can be shown that this curve lies in a plane. Assuming the curve lies in a plane, show that it is a circle centered at the origin with radius \(R\) provided \(a^{2}+c^{2}+e^{2}=b^{2}+d^{2}+f^{2}=R^{2}\) and \(a b+c d+e f=0\).

A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{array}{l} \mathbf{r}(t)=\langle 4+t,-2 t, 1+3 t\rangle ;\\\ \mathbf{R}(s)=\langle 1-7 s, 6+14 s, 4-21 s\rangle \end{array}$$

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and fare real numbers. It can be shown that this curve lies in a plane. Graph the following curve and describe it. $$\begin{aligned} \mathbf{r}(t)=&(2 \cos t+2 \sin t) \mathbf{i}+(-\cos t+2 \sin t) \mathbf{j} \\\ &+(\cos t-2 \sin t) \mathbf{k} \end{aligned}$$

A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{aligned} &\mathbf{r}(t)=\langle 3+4 t, 1-6 t, 4 t\rangle;\\\ &\mathbf{R}(s)=\langle-2 s, 5+3 s, 4-2 s\rangle \end{aligned}$$
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