Vector functions. Find r(t). How and why

Given r'(t)= (t^2)i + tj + (e^(2t))k and r(0) = -j+k, find r(t)

Given r'(t) = t² i + t j + e^(2t) k

I am first going to just use exp(x) = e^x since it makes my math easier to follow, secondly, I am going to use math vector notation since it is easier to follow than physics notation.

So the above is the same as:
r'(t) =


r(t) = ∫ r'(t) dt
r(t) = ∫ dt

I think perhaps you've never integrated a vector function before. You just integrate each component.
r(t) = <∫ t² dt,∫ t dt,∫ exp(2t) dt >
r(t) = <⅓ t³, .5 t², .5 exp(2t) > + C

Where C is a VECTOR constant.

Impose our initial condition:
r(0) = <0, 0, .5> + C = <0, -1, 1>
C = <0, -1, .5>

In math notation:
r(t) = <⅓ t³, .5 t² - 1, .5(exp(2t) + 1)>

I am switching back to physics notation.
r(t) = (⅓ t³) i + (.5 t² - 1)j + .5(exp(2t) + 1) k

Here,
r'(t)= (t^2)i + tj + (e^(2t))k and r(0) = -j+k, find r(t)

r(t) = (t^3/3)+(t^2/2)j +(e^2t/2)k + C,
for C, r(0) = -j+k, SO,

r(0) = 0 +0 +k/2 +C = -j+k,

C = -j +k/2,
,
r(t) = (t^3)i/3 +(t^2)-1)j/2 + (1+e^2t)k/2 >======< ANSWER