Questions: Find the general antiderivative for the function below; assume we have chosen an interval I on which the function is continuous. appropriate.) f(x)=(7+sqrt(x))^2/x F(x)= square x

To find the general antiderivative of the given function \( f(x) = \frac{(7+\sqrt{x})^{2}}{x} \), we first need to simplify the expression. We can expand the numerator and then divide each term by \( x \). After simplifying, we integrate each term separately to find the antiderivative.

Comenzamos con la función \( f(x) = \frac{(7+\sqrt{x})^{2}}{x} \). Al expandir el numerador, obtenemos: \[ f(x) = \frac{(7+\sqrt{x})^{2}}{x} = \frac{49 + 14\sqrt{x} + x}{x} = 1 + \frac{49}{x} + \frac{14}{\sqrt{x}} \]

Ahora, integramos cada término de la función simplificada: \[ F(x) = \int \left(1 + \frac{49}{x} + \frac{14}{\sqrt{x}}\right) \, dx \] Calculando cada integral, obtenemos: \[ F(x) = x + 49 \log(x) + 28 \sqrt{x} + C \] donde \( C \) es la constante de integración.

La antiderivada general de la función es: \[ \boxed{F(x) = x + 49 \log(x) + 28 \sqrt{x} + C} \]