Find the family of antiderivatives of

*d*

*dt*

*r*(*t*) = *t*3, *t* − 9, *t*2.

Given the vector-valued function \(\mathbf{r}(t) = \langle t^3, t - 9, t^2 \rangle\), we want to find the family of antiderivatives (or indefinite integrals) of \(\mathbf{r}(t)\).

### Solution

To find the antiderivative of a vector-valued function, we integrate each component of the vector with respect to \(t\). Let’s find the antiderivative of each component:

1. **First component**: \( t^3 \)
   
   \[
   \int t^3 \, dt = \frac{t^4}{4} + C_1
   \]

2. **Second component**: \( t - 9 \)
   
   \[
   \int (t - 9) \, dt = \frac{t^2}{2} - 9t + C_2
   \]

3. **Third component**: \( t^2 \)
   
   \[
   \int t^2 \, dt = \frac{t^3}{3} + C_3
   \]

Combining these results, the family of antiderivatives of \(\mathbf{r}(t)\) is:

\[
\int \mathbf{r}(t) \, dt = \left\langle \frac{t^4}{4} + C_1, \frac{t^2}{2} - 9t + C_2, \frac{t^3}{3} + C_3 \right\rangle
\]

where \(C_1\), \(C_2\), and \(C_3\) are arbitrary constants of integration.

### Final Answer

\[
\mathbf{R}(t) = \left\langle \frac{t^4}{4} + C_1, \frac{t^2}{2} - 9t + C_2, \frac{t^3}{3} + C_3 \right\rangle
\]

Let me know if you need more details or have any questions!

### Related Questions:
1. How do you find the definite integral of a vector-valued function?
2. What is the geometric interpretation of the antiderivative of a vector-valued function?
3. How do constant vectors affect the antiderivative of a vector function?
4. How can the antiderivative be used to find the position function of an object given its velocity?
5. What are the applications of antiderivatives in physics and engineering?

**Tip**: When integrating vector functions, always integrate each component separately and don’t forget to add constants of integration for each component!

Find the family of antiderivatives of d/dt r(t) = t^3, t - 9, t^2.

Discover how to find the family of antiderivatives for the vector-valued function r(t) = <t^3, t - 9, t^2>. This guide explains step-by-step integration of each component of the vector function and explores key calculus concepts, such as indefinite integrals and the Fundamental Theorem of Calculus.

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