Integrale di $$$\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}$$$

Trova $$$\int \left(\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}\right)\, dy$$$.

Integra termine per termine:

$${\color{red}{\int{\left(\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}\right)d y}}} = {\color{red}{\left(- \int{\sqrt{y} d y} + \int{\frac{y^{\frac{3}{2}}}{3} d y}\right)}}$$

Applica la regola della potenza $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=\frac{1}{2}$$$:

$$\int{\frac{y^{\frac{3}{2}}}{3} d y} - {\color{red}{\int{\sqrt{y} d y}}}=\int{\frac{y^{\frac{3}{2}}}{3} d y} - {\color{red}{\int{y^{\frac{1}{2}} d y}}}=\int{\frac{y^{\frac{3}{2}}}{3} d y} - {\color{red}{\frac{y^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}=\int{\frac{y^{\frac{3}{2}}}{3} d y} - {\color{red}{\left(\frac{2 y^{\frac{3}{2}}}{3}\right)}}$$

Applica la regola del fattore costante $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ con $$$c=\frac{1}{3}$$$ e $$$f{\left(y \right)} = y^{\frac{3}{2}}$$$:

$$- \frac{2 y^{\frac{3}{2}}}{3} + {\color{red}{\int{\frac{y^{\frac{3}{2}}}{3} d y}}} = - \frac{2 y^{\frac{3}{2}}}{3} + {\color{red}{\left(\frac{\int{y^{\frac{3}{2}} d y}}{3}\right)}}$$

Applica la regola della potenza $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=\frac{3}{2}$$$:

$$- \frac{2 y^{\frac{3}{2}}}{3} + \frac{{\color{red}{\int{y^{\frac{3}{2}} d y}}}}{3}=- \frac{2 y^{\frac{3}{2}}}{3} + \frac{{\color{red}{\frac{y^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}}{3}=- \frac{2 y^{\frac{3}{2}}}{3} + \frac{{\color{red}{\left(\frac{2 y^{\frac{5}{2}}}{5}\right)}}}{3}$$

Pertanto,

$$\int{\left(\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}\right)d y} = \frac{2 y^{\frac{5}{2}}}{15} - \frac{2 y^{\frac{3}{2}}}{3}$$

Semplifica:

$$\int{\left(\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}\right)d y} = \frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15}$$

Aggiungi la costante di integrazione:

$$\int{\left(\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}\right)d y} = \frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15}+C$$

$$$\int \left(\frac{y^{\frac{3}{2}}}{3} - \sqrt{y}\right)\, dy = \frac{2 y^{\frac{3}{2}} \left(y - 5\right)}{15} + C$$$A