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.