Integral de $$$\left(3 x + 1\right)^{2}$$$

Halla $$$\int \left(3 x + 1\right)^{2}\, dx$$$.

Sea $$$u=3 x + 1$$$.

Entonces $$$du=\left(3 x + 1\right)^{\prime }dx = 3 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{3}$$$.

La integral puede reescribirse como

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

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{3}$$$ y $$$f{\left(u \right)} = u^{2}$$$:

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

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

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

Recordemos que $$$u=3 x + 1$$$:

$$\frac{{\color{red}{u}}^{3}}{9} = \frac{{\color{red}{\left(3 x + 1\right)}}^{3}}{9}$$

Por lo tanto,

$$\int{\left(3 x + 1\right)^{2} d x} = \frac{\left(3 x + 1\right)^{3}}{9}$$

Añade la constante de integración:

$$\int{\left(3 x + 1\right)^{2} d x} = \frac{\left(3 x + 1\right)^{3}}{9}+C$$

$$$\int \left(3 x + 1\right)^{2}\, dx = \frac{\left(3 x + 1\right)^{3}}{9} + C$$$A