Calculus: Early Transcendentals, Chapter 4, Review, Section Review, Problem 70

You need to evaluate f(u) using the antiderivative of the function f'(u), such that:
int f'(u) du = f(u) + c
int (u^2 + sqrt u)/u du = int (u^2)/u du + int (sqrt u)/u du
int (u^2 + sqrt u)/u du = int u du + int u^(1/2 - 1) du
int (u^2 + sqrt u)/u du = u^2/2 + (u^(1/2 - 1+1))/(1/2 - 1+1) + c
int (u^2 + sqrt u)/u du = u^2/2 + 2sqrt u + c
Hence, f(u) = u^2/2 + 2sqrt u + c
You need to evaluate the constant c, using the information f(1) = 3, such that:
f(1) = 1^2/2 + 2sqrt 1 + c
3 = 1/2 + 2 + c => c = 3 - 2 - 1/2 => c = 1 - 1/2 => c = 1/2
Hence, evaluating the function f under the given conditions yields f(u) = u^2/2 + 2sqrt u + 1/2.

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