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Chapter 12: Problem 5

How do you find the indefinite integral of \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?\)

Short Answer

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Answer: The indefinite integral of the vector function \(\mathbf{r}(t) = \langle f(t), g(t),h(t)\rangle\) is \(\int \mathbf{r}(t) dt = \langle F(t) + C_1, G(t) + C_2, H(t) + C_3 \rangle\), where \(F(t)\), \(G(t)\), and \(H(t)\) are the antiderivatives of \(f(t)\), \(g(t)\), and \(h(t)\) respectively, and \(C_1\), \(C_2\), and \(C_3\) are the constants of integration.

Step by step solution

01

Identify the components of the vector function

The vector function \(\mathbf{r}(t) = \langle f(t), g(t),h(t)\rangle\) has three scalar components: \(f(t)\), \(g(t)\), and \(h(t)\). We will find the indefinite integral of each component separately and then combine the results to determine the indefinite integral of \(\mathbf{r}(t)\).
02

Find the indefinite integral of each component

Calculate the indefinite integrals of \(f(t)\), \(g(t)\), and \(h(t)\) with respect to \(t\). We do this by integrating each component function separately: - \(\int f(t) dt = F(t) + C_1\), where \(F(t)\) is an antiderivative of \(f(t)\) and \(C_1\) is the constant of integration, - \(\int g(t) dt = G(t) + C_2\), where \(G(t)\) is an antiderivative of \(g(t)\) and \(C_2\) is the constant of integration, - \(\int h(t) dt = H(t) + C_3\), where \(H(t)\) is an antiderivative of \(h(t)\) and \(C_3\) is the constant of integration.
03

Combine the results

Combine the results from Step 2 to find the indefinite integral of the vector function. The indefinite integral of \(\mathbf{r}(t)\) is the vector function formed by the indefinite integrals of its components. Hence, the indefinite integral of \(\mathbf{r}(t)=\langle f(t), g(t),h(t)\rangle\) is given by: $$\int \mathbf{r}(t) dt = \langle F(t) + C_1, G(t) + C_2, H(t) + C_3 \rangle$$

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