Question

Find r(t) if r'(t) = t⁵ i + et j + 3te⁽³ᵗ⁾ k and r(0) = i + j + k.

Answer 1

To find r(t), we need to integrate r'(t) with respect to t using the given components. The resulting r(t) is (1/6) t⁶ i + e^t j + (1/3) e⁽³ᵗ⁾ (3t - 1) k.

Step-by-step explanation:

To find r(t), we need to integrate r'(t) with respect to t. Given r'(t) = t⁵ i + et j + 3te⁽³ᵗ⁾ k, we can integrate each component separately.
Integrating the first component, we get ∫ t⁵ dt = (1/6) t⁶ + C1.
Integrating the second component, we get ∫ e^t dt = e^t + C2.
Integrating the third component, we get ∫ 3te⁽³ᵗ⁾ dt = (1/3) e⁽³ᵗ⁾ (3t - 1) + C3.
Combining these results, we have r(t) = (1/6) t⁶ i + e^t j + (1/3) e⁽³ᵗ⁾ (3t - 1) k.