# `f'(u) = (u^2 + sqrt(u))/u, f(1) = 3` Find `f`.

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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.` **