To solve this problem, we need to evaluate the integral:

$\int \langle 3t^3, e^{2t}, \sin(3t) \rangle \, dt$

This is a vector-valued integral, and we can integrate each component of the vector separately.

1. The integral of the first component, $3t^3$, is: $\int 3t^3 \, dt = \frac{3t^4}{4} + C_1$

2. The integral of the second component, $e^{2t}$, is: $\int e^{2t} \, dt = \frac{e^{2t}}{2} + C_2$

3. The integral of the third component, $\sin(3t)$, is: $\int \sin(3t) \, dt = -\frac{\cos(3t)}{3} + C_3$

Thus, combining the results, the final vector integral is:

$r(t) = \left\langle \frac{3t^4}{4} + C_1, \frac{e^{2t}}{2} + C_2, -\frac{\cos(3t)}{3} + C_3 \right\rangle$

Comparing this with the given options:

• Option (a) is the correct answer.

Would you like more details or further assistance?

Here are some related questions:

1. How do we evaluate a definite integral for vector-valued functions?
2. What is the method for integrating exponential and trigonometric functions separately?
3. How can we confirm the constants of integration in problems like this?
4. What is the geometric interpretation of vector-valued integrals?
5. How can we differentiate vector-valued functions?

Tip: Always remember that each component of a vector function is integrated separately.