Final answer:
To find r(t), integrate r'(t) with respect to t and solve for the constants using the given conditions. The resulting function is r(t) = t³i + t⁷j + ½t²k.
Step-by-step explanation:
To find r(t), we need to integrate r'(t) with respect to t. Starting with r'(t) = 3t²i + 7t⁶j + tk, we integrate each component separately.
∫3t² dt = t³ + C1
∫7t⁶ dt = t⁷ + C2
∫t dt = ½t² + C3
Combining these results, we have r(t) = (t³ + C1)i + (t⁷ + C2)j + (½t² + C3)k.
To find the specific values of C1, C2, and C3, we use the given conditions r(1) = i + j. Plugging in t = 1, we get:
r(1) = (1³ + C1)i + (1⁷ + C2)j + (½(1)² + C3)k = (1 + C1)i + (1 + C2)j + (½ + C3)k
Since this must equal i + j, we can equate the components and solve for the constants:
C1 = 0, C2 = 0, C3 = 0. Therefore, r(t) = t³i + t⁷j + ½t²k.