Find the indefinite integral.

(6ti + j + k) dt

To find the indefinite integral of the vector function \((6t\mathbf{i} + \mathbf{j} + \mathbf{k}) \, dt\), we integrate each component of the vector with respect to \(t\).

The vector function can be broken into its components:

\[
6t\mathbf{i} + \mathbf{j} + \mathbf{k}
\]

1. **Integral of \(6t\mathbf{i}\):**
   \[
   \int 6t \, dt = 6 \cdot \frac{t^2}{2} = 3t^2
   \]
   So, the integral of \(6t\mathbf{i}\) is \(3t^2\mathbf{i}\).

2. **Integral of \(\mathbf{j}\):**
   \[
   \int 1 \, dt = t
   \]
   So, the integral of \(\mathbf{j}\) is \(t\mathbf{j}\).

3. **Integral of \(\mathbf{k}\):**
   \[
   \int 1 \, dt = t
   \]
   So, the integral of \(\mathbf{k}\) is \(t\mathbf{k}\).

### Final result:

\[
\int (6t\mathbf{i} + \mathbf{j} + \mathbf{k}) \, dt = 3t^2\mathbf{i} + t\mathbf{j} + t\mathbf{k} + \mathbf{C}
\]

where \(\mathbf{C}\) is the constant of integration, a vector \((C_1\mathbf{i} + C_2\mathbf{j} + C_3\mathbf{k})\).

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Would you like more details on any of the steps?

Here are 5 related questions you can explore:
1. How do you find the indefinite integral of a vector function?
2. What is the difference between definite and indefinite integrals in vector calculus?
3. How do constants of integration work in vector calculus?
4. How would you compute a line integral for a vector field?
5. What is the geometric interpretation of integrating a vector function?

**Tip:** When integrating vector functions, treat each component separately, integrating them as scalar functions of the variable.

Find the indefinite integral. (6ti + j + k) dt

Find the indefinite integral of the vector function (6ti + j + k) dt. This detailed solution demonstrates component-wise integration, covering concepts from vector calculus and fundamental integration rules. Suitable for advanced high school and college-level students.

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