Final answer:
To find r(t), we need to integrate r'(t) with respect to t. The i, j and k components of r'(t) are integrated separately to find the corresponding components of r(t). The constant of integration is determined by substituting the known value of r(1) into the equation.
Step-by-step explanation:
The question asks us to find r(t) given that r'(t) = 5t^4i + 6t^5j + tK and r(1) = i + j.
To find r(t), we need to integrate r'(t) with respect to t. So, let's integrate each component separately:
For the i component, we integrate 5t^4 with respect to t and get (t^5)/5.
For the j component, we integrate 6t^5 with respect to t and get (t^6)/6.
For the k component, we integrate t with respect to t and get (t^2)/2.
Therefore, r(t) = (t^5)/5 * i + (t^6)/6 * j + (t^2)/2 * k + C, where C is a constant of integration. Since we know that r(1) = i + j, we can substitute t = 1 into the equation and solve for C.
r(1) = (1^5)/5 * i + (1^6)/6 * j + (1^2)/2 * k + C = i + j
1/5 + 1/6 + 1/2 + C = 1 + 1
C = 8/30 = 4/15
Therefore, the final equation for r(t) is r(t) = (t^5)/5 * i + (t^6)/6 * j + (t^2)/2 * k + 4/15.