Questions: Calcular la integral indefinida. [ int fracx-5sqrtx dx ]

To solve the indefinite integral \(\int \frac{x-5}{\sqrt{x}} \, dx\), we can first simplify the integrand by splitting it into two separate fractions and then integrating each term separately.

Comenzamos con la integral indefinida: \[ \int \frac{x-5}{\sqrt{x}} \, dx \] Podemos descomponer el integrando en dos términos: \[ \int \left( \frac{x}{\sqrt{x}} - \frac{5}{\sqrt{x}} \right) \, dx = \int \left( x^{1/2} - 5x^{-1/2} \right) \, dx \]

Ahora integramos cada término por separado: \[ \int x^{1/2} \, dx = \frac{2}{3} x^{3/2} \] \[ \int -5x^{-1/2} \, dx = -10\sqrt{x} \]

Sumamos los resultados de las integraciones: \[ \int \frac{x-5}{\sqrt{x}} \, dx = \frac{2}{3} x^{3/2} - 10\sqrt{x} + C \] donde \(C\) es la constante de integración.

\[ \boxed{\int \frac{x-5}{\sqrt{x}} \, dx = \frac{2}{3} x^{3/2} - 10\sqrt{x} + C} \]