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Chapter 12: Problem 48

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle 5 t^{-4}-t^{2}, t^{6}-4 t^{3}, 2 / t\right\rangle$$

Short Answer

Expert verified
Question: Find the indefinite integral of the given vector-valued function: $$\mathbf{r}(t)=\left\langle 5 t^{-4}-t^{2}, t^{6}-4 t^{3}, 2 / t\right\rangle$$ Answer: When you integrate the components of the given vector-valued function with respect to t, you will get: $$\int{(\mathbf{r}(t))}dt=\left\langle -\frac{5}{3}t^{-3} -\frac{1}{3}t^3 + C_1, \frac{1}{7}t^7 - t^4 + C_2, 2\ln{|t|} + C_3\right\rangle$$

Step by step solution

01

1. Observe the given function

The given function is a vector-valued function with three components. The function is given by: $$\mathbf{r}(t)=\left\langle 5 t^{-4}-t^{2}, t^{6}-4 t^{3}, 2 / t\right\rangle$$
02

2. Integrate the first component

To find the indefinite integral of the first component, we will integrate \((5t^{-4}-t^2)\) with respect to t: $$\int(5t^{-4}-t^2)dt = \int(5t^{-4})dt-\int(t^2)dt = 5\int(t^{-4})dt-\int(t^2)dt$$ Now we integrate each term: $$= -\frac{5}{3}t^{-3} -\frac{1}{3}t^3 + C_1$$ Where \(C_1\) is the constant of integration for the first component.
03

3. Integrate the second component

To find the indefinite integral of the second component, we will integrate \((t^{6}-4t^{3})\) with respect to t: $$\int(t^6 - 4t^3)dt = \int(t^6)dt - 4\int(t^3)dt$$ Now we integrate each term: $$= \frac{1}{7}t^7 - \frac{4}{4}t^4 + C_2$$ Where \(C_2\) is the constant of integration for the second component.
04

4. Integrate the third component

To find the indefinite integral of the third component, we will integrate \((2/t)\) with respect to t: $$\int(\frac{2}{t})dt=2\int(\frac{1}{t})dt$$ Now we integrate this term: $$= 2\ln{|t|} + C_3$$ Where \(C_3\) is the constant of integration for the third component.
05

5. Combine the integrated components

The indefinite integral of the given vector-valued function is the vector formed by combining the integrated components. Therefore, the integral of \(\mathbf{r}(t)\) is: $$\int{(\mathbf{r}(t))}dt=\left\langle -\frac{5}{3}t^{-3} -\frac{1}{3}t^3 + C_1, \frac{1}{7}t^7 - t^4 + C_2, 2\ln{|t|} + C_3\right\rangle$$

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