find f(t) from the following information f\'(t)=t i +t(1+t)^(-1/2)j + te^t k F(0

Question

                    find f(t) from the following information                 

                    f'(t)=t i +t(1+t)^(-1/2)j + te^t k F(0)= i+2j+3k
                

                    I know that f(t)=(t^2)/2 i +(t^2+1)^(1/2) j +e^t (t-1)k but i dont know what the constant vector would be. would it just be i+2j+3k? and if so do you just add                     it to the end of do you simplify it by adding it to the other three vector?
                

Explanation / Answer

by integrating f'(t)=t i +t(1+t)^(-1/2)j + te^t k

we get

F(t)= ( (t^2)/2 + a) i + ( (t^2+1)^(1/2) + b) j +( e^t (t-1) + c) k

now put initial condition F(0)= i+2j+3k , t = 0

i+2j+3k = ( 0+ a) i + ( (0+1)^(1/2) + b) j +( (0-1) + c) k

i+2j+3k = a i + ( 1 + b) j +( -1 + c) k

compare like terms

a = 1

1 + b = 2 ==> b = 1

-1 + c = 3 ==> c = 4

so,

F(t)= ( (t^2)/2 + 1) i + ( (t^2+1)^(1/2) + 1) j +( e^t (t-1) + 4) k

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