Integrale di $$$\frac{x}{\sqrt{4 - 5 x}}$$$

Trova $$$\int \frac{x}{\sqrt{4 - 5 x}}\, dx$$$.

Sia $$$u=4 - 5 x$$$.

Quindi $$$du=\left(4 - 5 x\right)^{\prime }dx = - 5 dx$$$ (i passaggi si possono vedere »), e si ha che $$$dx = - \frac{du}{5}$$$.

L'integrale diventa

$${\color{red}{\int{\frac{x}{\sqrt{4 - 5 x}} d x}}} = {\color{red}{\int{\frac{u - 4}{25 \sqrt{u}} d u}}}$$

Applica la regola del fattore costante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{25}$$$ e $$$f{\left(u \right)} = \frac{u - 4}{\sqrt{u}}$$$:

$${\color{red}{\int{\frac{u - 4}{25 \sqrt{u}} d u}}} = {\color{red}{\left(\frac{\int{\frac{u - 4}{\sqrt{u}} d u}}{25}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{\frac{u - 4}{\sqrt{u}} d u}}}}{25} = \frac{{\color{red}{\int{\left(\sqrt{u} - \frac{4}{\sqrt{u}}\right)d u}}}}{25}$$

Integra termine per termine:

$$\frac{{\color{red}{\int{\left(\sqrt{u} - \frac{4}{\sqrt{u}}\right)d u}}}}{25} = \frac{{\color{red}{\left(- \int{\frac{4}{\sqrt{u}} d u} + \int{\sqrt{u} d u}\right)}}}{25}$$

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

$$- \frac{\int{\frac{4}{\sqrt{u}} d u}}{25} + \frac{{\color{red}{\int{\sqrt{u} d u}}}}{25}=- \frac{\int{\frac{4}{\sqrt{u}} d u}}{25} + \frac{{\color{red}{\int{u^{\frac{1}{2}} d u}}}}{25}=- \frac{\int{\frac{4}{\sqrt{u}} d u}}{25} + \frac{{\color{red}{\frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}}{25}=- \frac{\int{\frac{4}{\sqrt{u}} d u}}{25} + \frac{{\color{red}{\left(\frac{2 u^{\frac{3}{2}}}{3}\right)}}}{25}$$

Applica la regola del fattore costante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=4$$$ e $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$:

$$\frac{2 u^{\frac{3}{2}}}{75} - \frac{{\color{red}{\int{\frac{4}{\sqrt{u}} d u}}}}{25} = \frac{2 u^{\frac{3}{2}}}{75} - \frac{{\color{red}{\left(4 \int{\frac{1}{\sqrt{u}} d u}\right)}}}{25}$$

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

$$\frac{2 u^{\frac{3}{2}}}{75} - \frac{4 {\color{red}{\int{\frac{1}{\sqrt{u}} d u}}}}{25}=\frac{2 u^{\frac{3}{2}}}{75} - \frac{4 {\color{red}{\int{u^{- \frac{1}{2}} d u}}}}{25}=\frac{2 u^{\frac{3}{2}}}{75} - \frac{4 {\color{red}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{25}=\frac{2 u^{\frac{3}{2}}}{75} - \frac{4 {\color{red}{\left(2 u^{\frac{1}{2}}\right)}}}{25}=\frac{2 u^{\frac{3}{2}}}{75} - \frac{4 {\color{red}{\left(2 \sqrt{u}\right)}}}{25}$$

Ricordiamo che $$$u=4 - 5 x$$$:

$$- \frac{8 \sqrt{{\color{red}{u}}}}{25} + \frac{2 {\color{red}{u}}^{\frac{3}{2}}}{75} = - \frac{8 \sqrt{{\color{red}{\left(4 - 5 x\right)}}}}{25} + \frac{2 {\color{red}{\left(4 - 5 x\right)}}^{\frac{3}{2}}}{75}$$

Pertanto,

$$\int{\frac{x}{\sqrt{4 - 5 x}} d x} = \frac{2 \left(4 - 5 x\right)^{\frac{3}{2}}}{75} - \frac{8 \sqrt{4 - 5 x}}{25}$$

Semplifica:

$$\int{\frac{x}{\sqrt{4 - 5 x}} d x} = \frac{2 \sqrt{4 - 5 x} \left(- 5 x - 8\right)}{75}$$

Aggiungi la costante di integrazione:

$$\int{\frac{x}{\sqrt{4 - 5 x}} d x} = \frac{2 \sqrt{4 - 5 x} \left(- 5 x - 8\right)}{75}+C$$

$$$\int \frac{x}{\sqrt{4 - 5 x}}\, dx = \frac{2 \sqrt{4 - 5 x} \left(- 5 x - 8\right)}{75} + C$$$A