can someone help me resolving this 2/u + u, f(1)=5 it's asking me to find the specific antiderivative.?

Question

This is business calculus.

Answer 1

f(u) = u + 2u^-1
F(u) = u²/2 + 2 ln |u| + C
F(1) = 1/2 + 2 ln |1| + C = 5
1/2 + 0 + C = 5
C = 4.5
so F(u) = u²/2 + 2 ln |u| + 4.5

Answer 2

It is given that f'(u) = 2/u + u

integrating both sides

f(u) = 2ln(u) + u^2/2 + c

f(1) = 2ln(1) + 1/2 + c = c + 1/2

so c + 1/2 = 5 , since f(1) = 5

c = 5-1/2 = 9/2

f(u) = 2ln(u) + u^2/2 + 9/2

f(u) = (1/2)[4ln(u) + u^2 + 9]