Find the most general antiderivative of the function. Use C for any needed constant.
f(u) = (u^4 + 7 sqrt(u)) / u^2
F(u) =
5,8,912,13,14,14,16,20
To find the most general antiderivative of the function f(u) = (u^4 + 7 sqrt(u)) / u^2, we can use the power rule and reverse the process of differentiation.
Step 1: Divide u^4 by u^2:
(u^4 + 7 sqrt(u)) / u^2 = u^2 + 7 u^(-1/2)
Step 2: Apply the power rule to the first term:
∫ u^2 du = (u^3)/3
Step 3: Apply the power rule to the second term:
∫ 7 u^(-1/2) du = 7 ∫ u^(-1/2) du
Step 4: Apply the power rule to the last term:
7 ∫ u^(-1/2) du = 7 * (2 u^(1/2))
Step 5: Simplify the expression:
7 * (2 u^(1/2)) = 14 u^(1/2)
Step 6: Combine the terms:
F(u) = (u^3)/3 + 14 u^(1/2) + C
Therefore, the most general antiderivative of the function f(u) = (u^4 + 7 sqrt(u)) / u^2 is F(u) = (u^3)/3 + 14 u^(1/2) + C, where C is a constant of integration.