Find r(t) if r′(t) = t i + et j + tet k and r(0) = i + j + k

To find \( r(t) \), we need to integrate \( r'(t) \) and then apply the initial condition \( r(0) = \mathbf{i} + \mathbf{j} + \mathbf{k} \).

Given:
\[
r'(t) = t \mathbf{i} + e^t \mathbf{j} + t e^t \mathbf{k}
\]

We will integrate each component with respect to \( t \):

1. **For the \( \mathbf{i} \) component:**
\[
\int t \, dt = \frac{t^2}{2}
\]

2. **For the \( \mathbf{j} \) component:**
\[
\int e^t \, dt = e^t
\]

3. **For the \( \mathbf{k} \) component:**
\[
\int t e^t \, dt
\]
We can use integration by parts for this:
\[
u = t, \quad dv = e^t dt \quad \Rightarrow \quad du = dt, \quad v = e^t
\]
Then,
\[
\int t e^t \, dt = t e^t - \int e^t \, dt = t e^t - e^t = (t - 1)e^t
\]

So, integrating \( r'(t) \), we get:
\[
r(t) = \frac{t^2}{2} \mathbf{i} + e^t \mathbf{j} + (t - 1) e^t \mathbf{k} + \mathbf{C}
\]
where \( \mathbf{C} \) is the constant vector of integration.

Now, apply the initial condition \( r(0) = \mathbf{i} + \mathbf{j} + \mathbf{k} \):
\[
r(0) = \frac{0^2}{2} \mathbf{i} + e^0 \mathbf{j} + (0 - 1) e^0 \mathbf{k} + \mathbf{C} = \mathbf{0} \mathbf{i} + 1 \mathbf{j} - 1 \mathbf{k} + \mathbf{C}
\]
\[
r(0) = \mathbf{C} + \mathbf{j} - \mathbf{k}
\]
Since \( r(0) = \mathbf{i} + \mathbf{j} + \mathbf{k} \), we have:
\[
\mathbf{C} + \mathbf{j} - \mathbf{k} = \mathbf{i} + \mathbf{j} + \mathbf{k}
\]
This gives:
\[
\mathbf{C} = \mathbf{i} + 0 \mathbf{j} + 2 \mathbf{k}
\]

Thus, the constant vector is \( \mathbf{C} = \mathbf{i} + 2 \mathbf{k} \).

Finally, the vector function \( r(t) \) is:
\[
r(t) = \frac{t^2}{2} \mathbf{i} + e^t \mathbf{j} + (t - 1) e^t \mathbf{k} + \mathbf{i} + 2 \mathbf{k}
\]
Simplifying:
\[
r(t) = \left( \frac{t^2}{2} + 1 \right) \mathbf{i} + e^t \mathbf{j} + \left( (t - 1) e^t + 2 \right) \mathbf{k}
\]

This is the final solution for \( r(t) \).

Would you like further details on any step?

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**Related questions:**
1. What is the method of integration by parts, and how does it apply in this example?
2. How do you apply initial conditions to solve for constants in vector-valued functions?
3. Can you explain how to simplify vector functions after integration?
4. What is the significance of the constant vector \( \mathbf{C} \) in vector calculus?
5. How does the exponential function behave in integrations involving products like \( t e^t \)?

**Tip:** When solving vector functions, treat each component separately and integrate accordingly.

Learn how to find the vector function r(t) when r′(t) = t i + e^t j + t e^t k and the initial condition r(0) = i + j + k. This step-by-step solution covers integration by parts, applying initial conditions, and simplifying vector-valued functions. Suitable for college-level calculus students.

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