Integral de $$$x \sqrt{x + 1}$$$

Halla $$$\int x \sqrt{x + 1}\, dx$$$.

Sea $$$u=x + 1$$$.

Entonces $$$du=\left(x + 1\right)^{\prime }dx = 1 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = du$$$.

La integral se convierte en

$${\color{red}{\int{x \sqrt{x + 1} d x}}} = {\color{red}{\int{\sqrt{u} \left(u - 1\right) d u}}}$$

Expand the expression:

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

Integra término a término:

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

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=\frac{3}{2}$$$:

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

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=\frac{1}{2}$$$:

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

Recordemos que $$$u=x + 1$$$:

$$- \frac{2 {\color{red}{u}}^{\frac{3}{2}}}{3} + \frac{2 {\color{red}{u}}^{\frac{5}{2}}}{5} = - \frac{2 {\color{red}{\left(x + 1\right)}}^{\frac{3}{2}}}{3} + \frac{2 {\color{red}{\left(x + 1\right)}}^{\frac{5}{2}}}{5}$$

Por lo tanto,

$$\int{x \sqrt{x + 1} d x} = \frac{2 \left(x + 1\right)^{\frac{5}{2}}}{5} - \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3}$$

Simplificar:

$$\int{x \sqrt{x + 1} d x} = \frac{2 \left(x + 1\right)^{\frac{3}{2}} \left(3 x - 2\right)}{15}$$

Añade la constante de integración:

$$\int{x \sqrt{x + 1} d x} = \frac{2 \left(x + 1\right)^{\frac{3}{2}} \left(3 x - 2\right)}{15}+C$$

$$$\int x \sqrt{x + 1}\, dx = \frac{2 \left(x + 1\right)^{\frac{3}{2}} \left(3 x - 2\right)}{15} + C$$$A